Abstract

Ž . Let p be a prime and let S be a finite p-group. Define d S to be the < < Ž . maximum of A as A ranges over the abelian subgroups of S, and A S to < < Ž . Ž . be the set of all abelian subgroups A for which A s d S . Let J S be Ž . the subgroup of S generated by A S . w x Ž . In 1964, Thompson T2 introduced a subgroup similar to J S and used it to obtain an improved version of his first normal p-complement theorem w x Ž . T1 . Since then, several variants of J S have been used to obtain related results. Most of their proofs have required a further result of Thompson, w x the Replacement Theorem Gor, p. 273; HB, III, p. 21 . Ž . Suppose A g A S , B is an abelian subgroup of S normalized by A, and Ž . B does not normalize A. Then there exists A* g A S such that

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