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Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups

Riccardo Aragona

Abstract

We show that if $G$ is a finite group whose Sylow $2$-subgroups are wreathed, then the intersection $\Outc(G) \cap \OutCol(G)$ has odd order, where $\Outc(G)$ and $\OutCol(G)$ denote the class-preserving and Coleman outer automorphism groups, respectively. This implies that $G$ satisfies the normalizer problem for its integral group ring. Combined with earlier work on the dihedral and semidihedral cases, this settles the question for all three families of $2$-groups of $2$-rank two classified by Gorenstein--Walter and Alperin--Brauer--Gorenstein.

Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups

Abstract

We show that if is a finite group whose Sylow -subgroups are wreathed, then the intersection has odd order, where and denote the class-preserving and Coleman outer automorphism groups, respectively. This implies that satisfies the normalizer problem for its integral group ring. Combined with earlier work on the dihedral and semidihedral cases, this settles the question for all three families of -groups of -rank two classified by Gorenstein--Walter and Alperin--Brauer--Gorenstein.
Paper Structure (5 sections, 17 theorems, 8 equations)

This paper contains 5 sections, 17 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wreathed 2-groups
  4. General results on Coleman automorphisms
  5. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

If $G$ is a finite group with wreathed Sylow $2$-subgroups, then $\operatorname{Out}_c(G) \cap \operatorname{Out}_{\mathrm{Col}}(G)$ is of odd order. In particular, $G$ satisfies the normalizer problem.

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6: Hertweck2001a, HK2002
  • Lemma 2.7: Hertweck2001a
  • ...and 23 more