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

TL;DR

This paper proves that finite groups with wreathed Sylow 2-subgroups have a specific automorphism intersection of odd order, resolving the normalizer problem for these groups and completing classifications for certain 2-group families.

Contribution

It establishes the odd order of the intersection of class-preserving and Coleman outer automorphisms for groups with wreathed Sylow 2-subgroups, completing the classification for all relevant 2-group families.

Findings

The intersection of automorphism groups has odd order.

The normalizer problem is satisfied for these groups.

Completes classification for all 2-rank two 2-groups.

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.