An extension of the Glauberman ZJ-theorem

Kızmaz, M. Yasi̇r

doi:10.1142/s0218196721500065

Cited by 2 publications

(1 citation statement)

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“…This includes (vi) and removed Glauberman's hypothesis that class(B) ≤ 2. Kızmaz [6] independently adapted Isaacs' arguments from [5] and proved his own generalization of the ZJ Theorem (on which our theorem 1.6 is modeled). This includes (vii), although he only stated the i = 1 case.…”

Section: Replacementmentioning

confidence: 99%

Variations on Glauberman's ZJ Theorem

Allcock¹

2020

Preprint

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We give a new proof of Glauberman's ZJ Theorem, in a form that clarifies the choices involved and offers more choices than classical treatments. In particular, we introduce two new ZJtype subgroups of a p-group S, that contain ZJ r (S) and ZJ o (S) respectively and are often strictly larger.Glauberman's ZJ Theorem is a basic technical tool in finite group theory. It plays a major role in the classification of simple groups having abelian or dihedral Sylow 2-subgroups. There are several versions of the theorem, depending on how one defines the Thompson subgroup. We develop the theorem in a way that clarifies the choices involved, and offers more choices than classical treatments.Writing S for a p-group, the following are new. First, we give an "axiomatic" version of the ZJ Theorem, theorem 1.1. Second, for p > 2 we construct ZJ-type groups ZJ lex (S) and ZJ olex (S), which contain ZJ r (S) resp. ZJ o (S) and can easily be larger. Third, we establish the "normalizers grow" property of the Thompson-Glauberman replacement process, and a consequence involving the Glauberman-Solomon group D * (S); see theorems 3.1(v) and 5.4.

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Section: Replacementmentioning

confidence: 99%