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Sastry automorphisms

E. Bombieri
Journal / Anthology

Journal of Algebra
Year: 2002
Volume: 257
Issue: 2
Page range: 222-243

Some time ago, N.S.N. Sastry [S] studied the question of the uniqueness of the embedding of the Ree groups 2F4 in the untwisted groups F4 over a field of characteristic 2. This led him to the problem of characterizing all automorphisms f of a field k of characteristic 2, with the following property: There are elements z = (z1, z2, z3, z4) in (k∗)4 such that: z1 = (z1z2)f ; (1) z1z−1 2 z3z4 = (z2z3)2f or (2I) z1z2z−1 3 z−1 4 = (z2z3)2f ; (2II) 1 + z1z−1 2 z−1 3 z4m1 +z2z−1 4 m1 + zm 3  = 1 + z1z−1 2 z−1 3 z4mf 1 + z2z−1 4 2mf 1 +(z3z4)mf , (3m) 1 + zm 4 31+ z2z−1 3 m1+ z3z−1 4 2m1+ z1z−1 2 z−1 3 z−1 4 m = 1 + z 4mf 4 1 + z2z−1 3 2mf 1 + z3z−1 4 mf 3 1+ z1z−1 2 z−1 3 z−1 4 mf  (4m) for m = 1, 2, 3, . . . . We shall refer to such an automorphism as a Sastry automorphism. E-mail address: eb@math.ias.edu. 0021-8693/02/$ see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0021-8693(02)00518-5 E. Bombieri / Journal of Algebra 257 (2002) 222243 223 A correspondence with Sastry ensued, which eventually led to the solution. The elimination of variables used is reminiscent of certain ideas which occur in the characterization of the Thompson automorphisms arising in the problem of uniqueness of the twisted Ree groups 2G2, see [T,B,E]. Thus, even if the embedding problem in the meantime may have been solved by less computational methods,1 it may be not inappropriate to present the result of this research here in an article dedicated to John Thompson. We note that we can always restrict k to the subfield k0 = F2(z) generated by the zi s over the field F2 of 2 elements. We say that a Sastry automorphism is of Type I or II according to whether (2I) or (2II) holds. It turns out that there are 22 families of solutions and 22 sporadic solutions over small finite fields, the largest of which is the Galois field

*Mathematics > Algebra > Group Theory

Ree groups