Some analogues of Glauberman’s 𝑍*-theorem

Let x be an element of prime order, p, in a finite group, G, and let P be a p-Sylow subgroup of G containing x. Say that x satisfies the unique conjugacy condition (u.c.c.) relative to G and the p-Sylow subgroup P if x is not conjugate in G to any other element of P. Let 02,(G) denote the largest normal subgroup of G of odd order. Define the subgroup Z*(G) by the equation Z*(G)/02'(G) = Z(G/021(G)), the center of G/02,(G). In the case that x is an involution satisfying the u.c.c., Glauberman [I] has shown that x lies in Z*(G). In particular, G is not nonabelian simple. Let O,'(G) denote the largest normal p'-subgroup of G. With an appropriate redefinition of Z*(G) by Z*(G)/0p,(G) =Z(G/Op,(G)), Glauberman [1] has asked whether an element x of odd prime order satisfying the u.c.c. lies in Z*(G). However, a resolution of this question (if one exists) appears to be difficult. Nonetheless, it is possible to make a compromise. If we strengthen the hypothesis of the proposition we can obtain analogues of Glauberman's Z*-theorem in which the conclusions are correspondingly stronger than xEZ*(G). (These appear as Corollary 2 and Theorem 2 below.) The theorems may be of interest in that they represent nonsimplicity criteria involving only local hypotheses and have relatively elementary proofs. The author wishes to thank Professor Glauberman for his suggestion to modify Corollary 2 to the version appearing as Theorem 1.