A 3-local characterization of U6(2) and Fi22

Abstract

The unitary group U 6 (2), often referred to as Fi 21 , and the sporadic simple group Fi 22 , discovered by Fischer [B. Fischer, Finite groups generated by 3-transpositions. I, Invent. Math. 13 (1971) 232-246 [6]], are characterized by specifying partial information about the structure of the normalizer of a non-trivial 3-central cyclic subgroup.

An Identification Theorem for Groups With Socle Psu

“…In earlier work [13] the first author proved the following result. Let G be a finite group, S be a Sylow 3-subgroup of G and Z = Z(S ).…”

“…Typically this will occur only when N G (Q) is not contained in H. If N H (Q) is nonsoluble and p is odd, Seidel has shown in his PhD thesis [22] that this cannot occur. In [17] the authors use the identification theorem presented in this paper together with further identifications [14,15,[18][19][20] to handle the more delicate analysis when p = 3 and N H (Q) is soluble. Far from these configurations not arising, the rule of thumb in these cases is that if it might happen then it does.…”

“…However, as Co 2 contains PSU 6 (2):2 as a subgroup of index 2300, these groups are intimately related. A 3-local identification of Co 2 can be found in [15].…”

“…Let G be a finite group, S be a Sylow 3-subgroup of G and Z = Z(S ). Assume that N G (Z) is similar to a 3-normalizer in PSU 6 (2) (see [13]). Then either G PSU 6 (2) or Z is weakly closed in S .…”

“…The fact that N G (J) is not contained in M is a consequence of the hypothesis that Z is not weakly closed in M. We find in Lemma 4.5 that N G (J)/J 2 × Sym(6) or Sym (6). With this information, after using a transfer theorem, we are able to apply [13], and do so in Theorem 4.7 to get that G PSU 6 (2) or PSU 6 (2):3 if N M (S )/S Dih (8). Thus from this stage on we assume that N M (S )/S 2 × Dih (8) and N M (J)/J 2 × Sym (6).…”