A characterization of saturated fusion systems over abelian 2-groups

TL;DR

This paper proves that in saturated fusion systems over abelian 2-groups, certain conjugacy conditions imply the group is abelian, simplifying existing proofs and revealing new conditions for abelian defect groups in blocks of finite groups.

Contribution

It generalizes a theorem of Camina--Herzog, providing a simplified proof and new criteria for abelian defect groups in block theory.

Findings

If every element is conjugate to an element in the center, the group is abelian.

All 2-blocks with major subsections have abelian defect groups.

2-blocks with symmetric stable centers have abelian defect groups.

Abstract

Given a saturated fusion system $\mathcal{F}$ over a $2$-group $S$, we prove that $S$ is abelian provided any element of $S$ is $\mathcal{F}$-conjugate to an element of $Z(S)$. This generalizes a Theorem of Camina--Herzog, leading to a significant simplification of its proof. More importantly, it follows that any $2$-block $B$ of a finite group has abelian defect groups if all $B$-subsections are major. Furthermore, every $2$-block with a symmetric stable center has abelian defect groups.