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Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

Robert M. Guralnick, Hung P. Tong-Viet, Gareth Tracey

Abstract

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$. In this paper, we first classify all finite groups $G$ which contains a weakly subnormal $p$-subgroup for some prime $p$. We then determine all finite groups containing a cyclic weakly subnormal $p$-subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements.

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

Abstract

A subgroup of a finite group is weakly subnormal in if is not subnormal in but it is subnormal in every proper overgroup of in . In this paper, we first classify all finite groups which contains a weakly subnormal -subgroup for some prime . We then determine all finite groups containing a cyclic weakly subnormal -subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements.
Paper Structure (7 sections, 30 theorems, 9 equations, 1 table)

This paper contains 7 sections, 30 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Weakly subnormal $p$-subgroups
  3. Reduction results for Baer-Suzuki type problems
  4. Applications to Baer-Suzuki type problems
  5. Glauberman's $Z_p^*$-theorem
  6. Orders of commutators and the open conjectures
  7. Nonlinear multiplicative irreducible characters

Key Result

Theorem 1

Let $p$ be a prime and let $G$ be a finite $p$-solvable group with $O_p(G)=1$. If $R$ is a weakly subnormal $p$-subgroup of $G$, then $G=QR$ where $Q$ is a special normal $q$-subgroup of $G$ for some prime $q \ne p$, $R$ centralizes $\Phi(Q)$, and $R$ acts irreducibly on $Q/\Phi(Q)$. In particular,

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Corollary 4
  • Remark 1.2
  • Theorem 5
  • Theorem 6
  • Conjecture 1
  • Theorem 7
  • ...and 53 more