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{{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0, 0, 1], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["3", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0, 0, 1], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["0", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(0\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(0\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 1, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["1", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 0.5, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["2", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0.559999, 0.370006, 0.599994], StripOnInput->False]} }, ColumnsEqual->False, GridBoxDividers->{ "Columns" -> {{1}}, "ColumnsIndexed" -> {2 -> Thickness[4]}, "Rows" -> {{1}}, "RowsIndexed" -> {2 -> Thickness[4]}}, GridBoxItemSize->{ "Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxItemStyle->{"Columns" -> {{ Directive["Panel", StandardForm]}}, "Rows" -> {{ Directive["Panel", StandardForm]}}}, GridBoxSpacings->{"Columns" -> {{ Offset[1]}}, "Rows" -> {{ Offset[1]}}}, RowsEqual->False], "Grid"]], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 2}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ Panel[ Grid[{{ Item[ Tooltip[ StandardForm["+"], "operator symbol"]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]]}, { Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]]}, { Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1], ItemStyle -> Directive["Panel", StandardForm]]], { Style["x", "Panel", Background -> None], Style["y", "Panel", Background -> None]}, {Left, Top}], Editable->False, Selectable->False], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ Panel[ Grid[{{ Item[ Tooltip[ StandardForm["+"], "operator symbol"]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", 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Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ 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{ StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0.559999, 0.370006, 0.599994], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["3", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], 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Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 0.5, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["2", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ 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FormBox["3", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, 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Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 1, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["1", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 0.5, 0], StripOnInput->False]}, { ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["3", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0, 0, 1], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["3", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0, 0, 1], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["0", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(0\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(0\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 1, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["1", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(4\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 0.5, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["2", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, 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ColumnsEqual->False, GridBoxDividers->{ "Columns" -> {{1}}, "ColumnsIndexed" -> {2 -> Thickness[4]}, "Rows" -> {{1}}, "RowsIndexed" -> {2 -> Thickness[4]}}, GridBoxItemSize->{ "Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxItemStyle->{"Columns" -> {{ Directive["Panel", StandardForm]}}, "Rows" -> {{ Directive["Panel", StandardForm]}}}, GridBoxSpacings->{"Columns" -> {{ Offset[1]}}, "Rows" -> {{ Offset[1]}}}, RowsEqual->False], "Grid"]], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 2}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ Panel[ Grid[{{ Item[ Tooltip[ StandardForm["+"], "operator symbol"]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]]}, { Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 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1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]]}, { Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1], ItemStyle -> Directive["Panel", StandardForm]]], { Style["x", "Panel", Background -> None], Style["y", "Panel", Background -> None]}, {Left, Top}], Editable->False, Selectable->False], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> 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RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[0], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(0\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(4\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> 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Style["\!\(0\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]]}, { Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[3], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 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GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, RowsEqual->False], "Column"]}], Row[{ Column[{ Tooltip[ Button[ Style["?", GrayLevel[1], Small], AbstractAlgebra`Master`Private`RightCosetsHelp[ AbstractAlgebra`Master`Private`groupoid[{0, 1, 2, 3, 4, 5, 6, 7}, Mod[# + #2, 8]& , {{}, {}, {}, {}, {}, { AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`FormatOperator -> False, AbstractAlgebra`Master`Generators -> {1}, AbstractAlgebra`Master`GroupoidDescription -> "Integers mod n under addition", AbstractAlgebra`Master`Private`GroupoidInputString -> "Z[8]", AbstractAlgebra`Master`GroupoidName -> "Z[8]"}}], {0, 4}], Background -> RGBColor[0.6, 0.4, 0.2], BaseStyle -> Small, ImageSize -> Tiny], "Click for further explanation of the image"], Panel[ Grid[{{ Item[ StandardForm["+"], Background -> GrayLevel[1]], Item[ StandardForm[0], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[4], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> 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Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 1, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox[ SuperscriptBox["Rot", "2"], StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(Rot\\^2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(Rot\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0.559999, 0.370006, 0.599994], StripOnInput->False]}, { ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox[ RowBox[{ SuperscriptBox["Rot", "2"], "**", "Ref"}], StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(Rot\\^2 ** Ref\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> 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StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox[ RowBox[{"Rot", "**", "Ref"}], StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(Rot ** Ref\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(Rot ** Ref\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** 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Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0, 0, 1], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox[ SuperscriptBox["Rot", "2"], StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(Rot\\^2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(Rot\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[0.559999, 0.370006, 0.599994], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["Rot", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(Rot\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(3\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(Rot\\^2\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 0.5, 0], StripOnInput->False], ItemBox[ TagBox[ TooltipBox[ TagBox[ FormBox["1", StandardForm], StandardForm, Editable->True], TagBox[ GridBox[{{ StyleBox["\"g\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"|g|\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\*SuperscriptBox[g, -1]\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}, { StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False], StyleBox["\"\\!\\(1\\)\"", RGBColor[0, 0, 1], 24, StripOnInput -> False]}}, GridBoxDividers -> { "Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, ColumnsEqual -> False, RowsEqual -> False], "Grid"]], Annotation[#, Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All], "Tooltip"]& ], Background->RGBColor[1, 1, 0], StripOnInput->False]} }, ColumnsEqual->False, GridBoxDividers->{ "Columns" -> {{1}}, "ColumnsIndexed" -> {2 -> Thickness[4]}, "Rows" -> {{1}}, "RowsIndexed" -> {2 -> Thickness[4]}}, GridBoxItemSize->{ "Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxItemStyle->{"Columns" -> {{ Directive[Small, StandardForm]}}, "Rows" -> {{ Directive[Small, StandardForm]}}}, GridBoxSpacings->{"Columns" -> {{ Offset[1]}}, "Rows" -> {{ Offset[1]}}}, RowsEqual->False], "Grid"]], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 2}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ Panel[ Grid[{{ Item[ Tooltip[ StandardForm["*"], "operator symbol"]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]]}, { Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, 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1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]]}, { Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]]}, { Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]]}, { Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]]}, { Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}, { Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1], ItemStyle -> Directive[Small, StandardForm]]], { Style["x", "Panel", Background -> None], Style["y", "Panel", Background -> None]}, {Left, Top}], Editable->False, Selectable->False], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ Panel[ Grid[{{ Item[ Tooltip[ StandardForm["*"], "operator symbol"]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]]}, { Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[1], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24], Style["\!\(1\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 1, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot^2], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Ref], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0, 0, 1]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ Tooltip[ StandardForm[ NonCommutativeMultiply[ AbstractAlgebra`Master`Rot^2, AbstractAlgebra`Master`Ref]], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24], Style["\!\(2\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2 ** Ref\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[0.250999, 0.878399, 0.815699]]}, { Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 0, 1], 24], Style["\*SuperscriptBox[g, -1]", RGBColor[0, 0, 1], 24]}, { Style["\!\(Rot\)", RGBColor[0, 0, 1], 24], Style["\!\(3\)", RGBColor[0, 0, 1], 24], Style["\!\(Rot\^2\)", RGBColor[0, 0, 1], 24]}}, Dividers -> All]], Background -> RGBColor[1, 0.5, 0]], Item[ Tooltip[ StandardForm[AbstractAlgebra`Master`Rot], Grid[{{ Style["g", RGBColor[0, 0, 1], 24], Style["|g|", RGBColor[0, 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AbstractAlgebra`Master`Textual, AbstractAlgebra`Master`ShowConvert -> False], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$1339$$, -1, "g \[Element] G"}, {-1, 0, 1}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}, {3,"\<\"Visualization\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$1340$$ = -1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$1340$$], -1, "g \[Element] G"}, {-1, 0, 1}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1340$1375$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {AbstractAlgebra`Master`Private`h$1340$$ = -1}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$1340$$, AbstractAlgebra`Master`Private`h$1340$1375$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> AbstractAlgebra`Master`InvertibleQ[ AbstractAlgebra`Master`Private`groupoid[{-1, 0, 1}, Plus, {{}, {}, {}, {}, {}, {}}], AbstractAlgebra`Master`Private`h$1340$$, Mode -> AbstractAlgebra`Master`Visual, AbstractAlgebra`Master`ShowConvert -> False], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$1340$$, -1, "g \[Element] G"}, {-1, 0, 1}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}}, 1, ImageSize->Automatic], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ TabView[{"Computation" -> Manipulate[ AbstractAlgebra`Master`InvertibleQ[ AbstractAlgebra`Master`Private`groupoid[{-1, 0, 1}, Plus, {{}, {}, {}, {}, {}, {}}], AbstractAlgebra`Master`Private`h$1338, Mode -> AbstractAlgebra`Master`Computational], {{ AbstractAlgebra`Master`Private`h$1338, -1, "g \[Element] G"}, {-1, 0, 1}}, ControlType -> RadioButtonBar], "Explanation" -> Manipulate[ AbstractAlgebra`Master`InvertibleQ[ AbstractAlgebra`Master`Private`groupoid[{-1, 0, 1}, Plus, {{}, {}, {}, {}, {}, {}}], AbstractAlgebra`Master`Private`h$1339, Mode -> AbstractAlgebra`Master`Textual, AbstractAlgebra`Master`ShowConvert -> False], {{ AbstractAlgebra`Master`Private`h$1339, -1, "g \[Element] G"}, {-1, 0, 1}}, ControlType -> RadioButtonBar], "Visualization" -> Manipulate[ AbstractAlgebra`Master`InvertibleQ[ AbstractAlgebra`Master`Private`groupoid[{-1, 0, 1}, Plus, {{}, {}, {}, {}, {}, {}}], AbstractAlgebra`Master`Private`h$1340, Mode -> AbstractAlgebra`Master`Visual, AbstractAlgebra`Master`ShowConvert -> False], {{ AbstractAlgebra`Master`Private`h$1340, -1, "g \[Element] G"}, {-1, 0, 1}}, ControlType -> RadioButtonBar]}, ImageSize -> Automatic], "Interacting with InvertibleQ[G, g]", Top], Editable->False, Selectable->False]], "Output", CellChangeTimes->{3.435012760910884*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"OrderOfElement", "[", RowBox[{ RowBox[{"D", "[", RowBox[{"4", ",", RowBox[{"Form", "\[Rule]", "Permutations"}]}], "]"}], ",", RowBox[{"{", RowBox[{"2", ",", "1", ",", "4", ",", "3"}], "}"}], ",", RowBox[{"Mode", "\[Rule]", "Interactive"}]}], "]"}]], "Input", CellChangeTimes->{{3.433807205484757*^9, 3.433807251182239*^9}, { 3.433807293130494*^9, 3.433807356757477*^9}}], Cell[BoxData[ InterpretationBox[GridBox[{ { TagBox["\<\"Interacting with OrderOfElement[G, g]\"\>", "Labeled", Editable->True, Selectable->True]}, { TagBox[ TagBox[ TabViewBox[{{1,"\<\"Computation\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$$], {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, { 2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}}, Typeset`size$$ = {13.5, {0., 15.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1411$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> { AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$$, AbstractAlgebra`Master`Private`h$1411$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular \ n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h$$, Mode -> AbstractAlgebra`Master`Computational], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$$, {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, { 4, 3, 2, 1}}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], ImageSizeCache->{745.5, {73.8125, 83.6875}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}, {2,"\<\"Explanation\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$$], {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, { 2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1428$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> { AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$$, AbstractAlgebra`Master`Private`h$1428$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular \ n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h$$, Mode -> AbstractAlgebra`Master`Textual], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$$, {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, { 4, 3, 2, 1}}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}, {3,"\<\"Visualization\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$$], {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, { 2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1445$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> { AbstractAlgebra`Master`Private`h$$ = {2, 1, 4, 3}}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$$, AbstractAlgebra`Master`Private`h$1445$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular \ n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h$$, Mode -> AbstractAlgebra`Master`Visual], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$$, {2, 1, 4, 3}, "g \[Element] G"}, {{1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, { 4, 3, 2, 1}}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}}, 1, ImageSize->Automatic], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ TabView[{"Computation" -> Manipulate[ AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, { 2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h, Mode -> AbstractAlgebra`Master`Computational], {{ AbstractAlgebra`Master`Private`h, {2, 1, 4, 3}, "g \[Element] G"}, {{ 1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, { 3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}, ControlType -> RadioButtonBar], "Explanation" -> Manipulate[ AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, { 2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h, Mode -> AbstractAlgebra`Master`Textual], {{ AbstractAlgebra`Master`Private`h, {2, 1, 4, 3}, "g \[Element] G"}, {{ 1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, { 3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}, ControlType -> RadioButtonBar], "Visualization" -> Manipulate[ AbstractAlgebra`Master`OrderOfElement[ AbstractAlgebra`Master`Private`groupoid[{{1, 2, 3, 4}, {1, 4, 3, 2}, { 2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, {3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}, Map[ Function[AbstractAlgebra`Master`Private`k, Part[#, AbstractAlgebra`Master`Private`k]], #2]& , {{}, {}, {}, \ {}, {}, {AbstractAlgebra`Master`SizeLimit -> 8, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Dihedral group - reflections and rotations of a regular n-gon", AbstractAlgebra`Master`WideElements -> True, AbstractAlgebra`Master`Private`GroupoidInputString -> "Dihedral[4]", AbstractAlgebra`Master`GroupoidName -> "D[4]"}}], AbstractAlgebra`Master`Private`h, Mode -> AbstractAlgebra`Master`Visual], {{ AbstractAlgebra`Master`Private`h, {2, 1, 4, 3}, "g \[Element] G"}, {{ 1, 2, 3, 4}, {1, 4, 3, 2}, {2, 1, 4, 3}, {2, 3, 4, 1}, {3, 2, 1, 4}, { 3, 4, 1, 2}, {4, 1, 2, 3}, {4, 3, 2, 1}}}, ControlType -> RadioButtonBar]}, ImageSize -> Automatic], "Interacting with OrderOfElement[G, g]", Top], Editable->False, Selectable->False]], "Output", CellChangeTimes->{3.435012762960106*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PreservesQ", "[", RowBox[{ RowBox[{"ZMap", "[", RowBox[{"4", ",", "2"}], "]"}], ",", RowBox[{"{", RowBox[{"2", ",", "3"}], "}"}], ",", RowBox[{"Mode", "\[Rule]", "Interactive"}]}], "]"}]], "Input", CellChangeTimes->{{3.433807634985114*^9, 3.433807637618554*^9}, { 3.433872164713936*^9, 3.433872181950935*^9}, 3.433893928773604*^9}], Cell[BoxData[ InterpretationBox[GridBox[{ { TagBox["\<\"Interacting with PreservesQ[f, {g, h}]\"\>", "Labeled", Editable->True, Selectable->True]}, { TagBox[ TagBox[ TabViewBox[{{1,"\<\"Computation\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$$ = {2, 3}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$$], {2, 3}, "{g, h} \[Subset] G"}, {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {3, 0}, {3, 1}, {3, 2}, {3, 3}}}}, Typeset`size$$ = {54., {1., 13.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1486$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {AbstractAlgebra`Master`Private`h$$ = {2, 3}}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$$, AbstractAlgebra`Master`Private`h$1486$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> AbstractAlgebra`Master`PreservesQ[ AbstractAlgebra`Master`Private`morphoid[Mod[#, 2]& , AbstractAlgebra`Master`Private`groupoid[{0, 1, 2, 3}, Mod[# + #2, 4]& , {{}, {}, {}, {}, {}, { AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`FormatOperator -> False, AbstractAlgebra`Master`Generators -> {1}, AbstractAlgebra`Master`GroupoidDescription -> "Integers mod n under addition", AbstractAlgebra`Master`Private`GroupoidInputString -> "Z[4]", AbstractAlgebra`Master`GroupoidName -> "Z[4]"}}], AbstractAlgebra`Master`Private`groupoid[{0, 1}, Mod[# + #2, 2]& , {{}, {}, {}, {}, {}, { AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`FormatOperator -> False, AbstractAlgebra`Master`Generators -> {1}, AbstractAlgebra`Master`GroupoidDescription -> "Integers mod n under addition", AbstractAlgebra`Master`Private`GroupoidInputString -> "Z[2]", AbstractAlgebra`Master`GroupoidName -> "Z[2]"}}], AbstractAlgebra`Master`Group, Function, AbstractAlgebra`Master`FormatFunction -> Rule, AbstractAlgebra`Master`Private`GeneratingRule -> {1 -> 1}, AbstractAlgebra`Master`Private`ConjugatingElement -> Null], AbstractAlgebra`Master`Private`h$$, Mode -> AbstractAlgebra`Master`Computational], "Specifications" :> {{{ AbstractAlgebra`Master`Private`h$$, {2, 3}, "{g, h} \[Subset] G"}, {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, { 3, 0}, {3, 1}, {3, 2}, {3, 3}}}}, "Options" :> {ControlType -> RadioButtonBar}, "DefaultOptions" :> {}], ImageSizeCache->{745.5, {73.8125, 83.6875}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]}, {2,"\<\"Visualization\"\>"-> TagBox[ StyleBox[ DynamicModuleBox[{AbstractAlgebra`Master`Private`h$$ = {2, 3}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[AbstractAlgebra`Master`Private`h$$], {2, 3}, "{g, h} \[Subset] G"}, {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {3, 0}, {3, 1}, {3, 2}, {3, 3}}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, AbstractAlgebra`Master`Private`h$1503$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {AbstractAlgebra`Master`Private`h$$ = {2, 3}}, "ControllerVariables" :> { Hold[ AbstractAlgebra`Master`Private`h$$, AbstractAlgebra`Master`Private`h$1503$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 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AbstractAlgebra`Master`ShowHelp -> AbstractAlgebra`Master`Private`showhelp$1569], {{ AbstractAlgebra`Master`Private`k, 4, "n"}, 2, 20, 1, Appearance -> "Open"}], "Visualization" -> Manipulate[ AbstractAlgebra`Master`Dihedral[ AbstractAlgebra`Master`Private`k, Mode -> AbstractAlgebra`Master`Visual, AbstractAlgebra`Master`ShowConvert -> False, AbstractAlgebra`Master`ShowHelp -> AbstractAlgebra`Master`Private`showhelp$1576], {{ AbstractAlgebra`Master`Private`k, 4, "n"}, 2, 20, 1, Appearance -> "Open"}]}, ImageSize -> Automatic], "Interacting with Dihedral[n]", Top], Editable->False, Selectable->False]}}, 1, ImageSize->Automatic], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ TabView[{ "Computation" -> 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"Explanation" -> "One way of thinking of the Dihedral group with index 4 is to consider \ the regular 4-gon. The collection of symmetries for this figure makes up this \ group, with the elements being represented (symbolically) as the elements \ generated by the smallest rotation and any reflection.", "Visualization" -> Column[{ Graphics[{{{ RGBColor[0.739998, 0.990005, 0.790002], Polygon[{{0.7071067811865475, 0.7071067811865475}, { 0.7071067811865475, -0.7071067811865475}, {-0.7071067811865475, \ -0.7071067811865475}, {-0.7071067811865475, 0.7071067811865475}}]}, { GrayLevel[0], Line[{{0.7071067811865475, 0.7071067811865475}, { 0.7071067811865475, -0.7071067811865475}, {-0.7071067811865475, \ -0.7071067811865475}, {-0.7071067811865475, 0.7071067811865475}, { 0.7071067811865475, 0.7071067811865475}}]}}, { Text[1, {0.8838834764831843, 0.8838834764831843}], Text[2, {0.8838834764831843, -0.8838834764831843}], Text[3, {-0.8838834764831843, -0.8838834764831843}], Text[4, {-0.8838834764831843, 0.8838834764831843}]}, {{ PointSize[0.045], RGBColor[1, 0, 0], Point[{0.7071067811865475, 0.7071067811865475}]}, { PointSize[0.045], RGBColor[0, 1, 0], Point[{0.7071067811865475, -0.7071067811865475}]}, { PointSize[0.045], RGBColor[0, 0, 1], Point[{-0.7071067811865475, -0.7071067811865475}]}, { PointSize[0.045], RGBColor[1, 1, 0], Point[{-0.7071067811865475, 0.7071067811865475}]}}}, {}], Grid[{{ Tooltip[ Button[ Style["?", GrayLevel[1], Small], AbstractAlgebra`Master`Private`DihedralHelp[4], Background -> RGBColor[0.6, 0.4, 0.2], BaseStyle -> Small, ImageSize -> Tiny], "Click for further explanation of the image"]}}, ItemSize -> All]}, Center], "Interactive" -> Labeled[ TabView[{"Computation" -> Manipulate[ AbstractAlgebra`Master`Dihedral[ AbstractAlgebra`Master`Private`k, Mode -> AbstractAlgebra`Master`Computational], {{ AbstractAlgebra`Master`Private`k, 4, "n"}, 2, 20, 1, Appearance -> "Open"}], "Explanation" -> Manipulate[ AbstractAlgebra`Master`Dihedral[ AbstractAlgebra`Master`Private`k, Mode -> AbstractAlgebra`Master`Textual, AbstractAlgebra`Master`ShowConvert -> False, AbstractAlgebra`Master`ShowHelp -> AbstractAlgebra`Master`Private`showhelp$1569], {{ AbstractAlgebra`Master`Private`k, 4, "n"}, 2, 20, 1, Appearance -> "Open"}], "Visualization" -> Manipulate[ AbstractAlgebra`Master`Dihedral[ AbstractAlgebra`Master`Private`k, Mode -> AbstractAlgebra`Master`Visual, AbstractAlgebra`Master`ShowConvert -> False, AbstractAlgebra`Master`ShowHelp -> AbstractAlgebra`Master`Private`showhelp$1576], {{ AbstractAlgebra`Master`Private`k, 4, "n"}, 2, 20, 1, Appearance -> "Open"}]}, ImageSize -> Automatic], "Interacting with Dihedral[n]", Top]}, ImageSize -> Automatic], "The dihedral group D[4]", Top], Editable->False, Selectable->False]], "Output", CellChangeTimes->{3.435012769756468*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SubgroupQ", "[", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "21"}], "}"}], ",", RowBox[{"U", "[", "22", "]"}], ",", RowBox[{"Mode", "\[Rule]", "All"}]}], "]"}]], "Input", CellChangeTimes->{{3.43380665383193*^9, 3.4338066666240997`*^9}, { 3.433806812073365*^9, 3.4338068558077097`*^9}, {3.433806908321823*^9, 3.433806919251732*^9}}], Cell[BoxData[ InterpretationBox[GridBox[{ { TagBox["\<\"Is \\!\\(TraditionalForm\\`\\({1, 21}\\)\\) a subgroup of \ U[22]?\"\>", "Labeled", Editable->True, Selectable->True]}, { TagBox[ TagBox[ TabViewBox[{{1,"\<\"Computation\"\>"->"True"}, { 2,"\<\"Explanation\"\>"->"\<\"By definition, we say a set H is a \ subgroup of the group G if H is a subset of G and it is itself a group under \ the operation inherited from G. 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StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}, { Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]]}, { Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]]}, { Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]]}, { Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]]}, { Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]]}, { Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1]]], { Style["x", "Panel", Background -> None], Style["y", "Panel", Background -> None]}, {Left, Top}], Editable->False, Selectable->False]} }, ColumnsEqual->False, GridBoxAlignment->{"Columns" -> {{Center}}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, RowsEqual->False], "Column"]}}, 1, ImageSize->Automatic], "Labeled", Editable->True, Selectable->True], "SkipImageSizeLevel"]} }, BaselinePosition->{2, 1}, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}], Labeled[ TabView[{ "Computation" -> True, "Explanation" -> "By definition, we say a set H is a subgroup of the group G if H is a \ subset of G and it is itself a group under the operation inherited from G. \ For finite groups, as is the case here, one only needs to show the closure of \ H under the operation of G (assuming, of course, that H is nonempty).\nSince \ \!\(TraditionalForm\`\({1, 21}\)\) is closed under the operation of U[22], \ then indeed this is a subgroup.\n", "Visualization 1" -> Column[{ Panel[ Grid[{{ Item[ StandardForm["*"], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[21], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]]}, { Item[ StandardForm[1], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[1], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[21], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]]}, { Item[ StandardForm[21], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[21], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[1], Background -> RGBColor[0.48627450980392156`, 0.9882352941176471, 0.]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]]}, { Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]]}, { Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]]}, { Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]]}, { Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]]}, { Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]]}, { Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]]}, { Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]]}, { Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[19], Background -> GrayLevel[1]], Item[ StandardForm[3], Background -> GrayLevel[1]], Item[ StandardForm[13], Background -> GrayLevel[1]], Item[ StandardForm[7], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> GrayLevel[1]], Item[ StandardForm[17], Background -> GrayLevel[1]], Item[ StandardForm[5], Background -> GrayLevel[1]], Item[ StandardForm[21], Background -> GrayLevel[1]], Item[ StandardForm[15], Background -> GrayLevel[1]], Item[ StandardForm[9], Background -> GrayLevel[1]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1]], {"x", "y"}, {Left, Top}], Grid[{{ Tooltip[ Button[ Style["?", GrayLevel[1], Small], AbstractAlgebra`Master`Private`SubgroupQHelp[{1, 21}, AbstractAlgebra`Master`Private`groupoid[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}, Mod[# #2, 22]& , {{}, {}, {}, {}, {}, { AbstractAlgebra`Master`FormatOperator -> False, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Integers less than n and relatively prime to n, under \ multiplication", AbstractAlgebra`Master`Private`GroupoidInputString -> "U[22]", AbstractAlgebra`Master`GroupoidName -> "U[22]"}}]], Background -> RGBColor[0.6, 0.4, 0.2], BaseStyle -> Small, ImageSize -> Tiny], "Click for further explanation of the image"]}}, ItemSize -> All]}, Center], "Visualization 2" -> Column[{ If[Show, Button["Click for explanation", AbstractAlgebra`Master`Private`SubgroupQ2Help[{1, 21}, AbstractAlgebra`Master`Private`groupoid[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}, Mod[# #2, 22]& , {{}, {}, {}, {}, {}, { AbstractAlgebra`Master`FormatOperator -> False, AbstractAlgebra`Master`IsAGroup -> True, AbstractAlgebra`Master`GroupoidDescription -> "Integers less than n and relatively prime to n, under \ multiplication", AbstractAlgebra`Master`Private`GroupoidInputString -> "U[22]", AbstractAlgebra`Master`GroupoidName -> "U[22]"}}]], Background -> Yellow, BaseStyle -> Small, ImageSize -> Automatic], ""], Panel[ Grid[{{ Item[ StandardForm["*"], Background -> GrayLevel[1]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]]}, { Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]]}, { Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]]}, { Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}, { Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]]}, { Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]]}, { Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]]}, { Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]]}, { Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]]}, { Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]]}, { Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[13], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[9], Background -> RGBColor[0.739998, 0.990005, 0.790002]], Item[ StandardForm[17], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[5], Background -> RGBColor[0.559999, 0.370006, 0.599994]], Item[ StandardForm[21], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[1], Background -> RGBColor[1, 1, 0]], Item[ StandardForm[3], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[19], Background -> RGBColor[1, 0.5, 0]], Item[ StandardForm[7], Background -> RGBColor[0, 0, 1]], Item[ StandardForm[15], Background -> RGBColor[0, 0, 1]]}}, Dividers -> {{1, 2 -> Thickness[4]}, {1, 2 -> Thickness[4]}}, ItemSize -> Full, Spacings -> Offset[1]], {"x", "y"}, {Left, Top}]}, Center]}, ImageSize -> Automatic], "Is \!\(TraditionalForm\`\({1, 21}\)\) a subgroup of U[22]?", Top], Editable->False, Selectable->False]], "Output", 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This link calls upon a preliminary version of MorphismInvestigator, one \ of five parts of AlgebraInvestigator. Here one can click one's way through \ setting up a morphism by matching a list of elements in the domain to \ elements in the codomain. This can be found (now or soon) at the same site of \ the AbstractAlgebra package: ", ButtonBox["http://www.central.edu/eaam", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.central.edu/eaam"], None}], "." }], "Text", CellChangeTimes->{{3.4338077085937357`*^9, 3.4338077309312143`*^9}, { 3.4338077797318983`*^9, 3.433807779751782*^9}, {3.4350124042399063`*^9, 3.43501254728258*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Version 6", "Subsection", CellChangeTimes->{{3.433804334894854*^9, 3.433804349064732*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FormMorphoidSetup", "[", RowBox[{ RowBox[{"D", "[", "4", "]"}], ",", RowBox[{"Z", "[", "8", "]"}], ",", RowBox[{"Mode", "\[Rule]", "Interactive"}]}], "]"}]], "Input", CellChangeTimes->{{3.433808248831465*^9, 3.433808277902128*^9}}], Cell[BoxData[ TagBox[GridBox[{ { InterpretationBox[ RowBox[{"\<\"Match elements for map: \"\>", "\[InvisibleSpace]", PopupMenuBox[Dynamic[ AbstractAlgebra`Master`Private`CandidateRules[1]], {1 -> 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