A characterization of the simple groups PSp(4,3) and PSp(6,2)
Denote by \(H_0\) the centralizer in \(\text{PSp}(4,3)\) of an element of order 3 in the center of a Sylow 3-subgroup, and denote by \(H^*_0\) the extended centralizer in \(\text{Aut}[\text{PSp}(4,3)]\). The following theorem is proved. Let \(G\) be a finite group containing an element \(\alpha\) of order 3 such that (a) \(C_G(\alpha)\) is isomorphic to \(H_o^*\); (b) \(C_G(\alpha)\) contains an elementary abelian subgroup \(M\) of order 27 which does not normalize any nontrivial 3'-subgroup of \(G\); (c) \(\alpha\) is conjugate to \(\alpha^{-1}\). Then one of the following holds: (i) \(G\) is isomorphic to \(H_o^*\); (ii) \(G\) is isomorphic to \(0_6^- (2)\); (iii) \(G\) is isomorphic to \(\text{Psp}(6,2)\). This complements a theorem of \textit{J. L.Hayden} [Canadian J. Math. 25, 539--553 (1973; Zbl 0267.20039)], which deals with the case that \(\alpha\) is not conjugate to \(\alpha6{-1}\). The terminology ``3-centralizer structure'' is introduced. Two finite groups are said to have the same 3-centralizer structure if they have the same number of conjugacy classes of elements of order 3 and the corresponding centralizers are isomorphic. It is proved that the simple groups \(\text{PSp}(4,3)\) and \(\text{PSp}(6,2)\) are characterized by their respective 3-centralizer structures.
scientific article; zbMATH DE number 3558063