An extension of the Glauberman ZJ-theorem

Abstract

Let [Formula: see text] be an odd prime and let [Formula: see text], [Formula: see text] and [Formula: see text] denote the three different versions of Thompson subgroups for a [Formula: see text]-group [Formula: see text]. In this paper, we first prove an extension of Glauberman’s replacement theorem [G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 (1968) 1101–1135, Theorem 4.1]. Second, we prove the following: Let [Formula: see text] be a [Formula: see text]-stable group and [Formula: see text]. Suppose that [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then [Formula: see text], [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text]. Third, we show the following: Let [Formula: see text] be a [Formula: see text]-free group and [Formula: see text]. If [Formula: see text] is a strongly closed subgroup in [Formula: see text], then the normalizers of the subgroups [Formula: see text], [Formula: see text] and [Formula: see text] control strong [Formula: see text]-fusion in [Formula: see text]. We also prove a similar result for a [Formula: see text]-stable and [Formula: see text]-constrained group. Finally, we give a [Formula: see text]-nilpotency criteria, which is an extension of Glauberman–Thompson [Formula: see text]-nilpotency theorem.