Blocks with transitive fusion systems

TL;DR

This paper classifies the defect groups of certain blocks in modular representation theory, showing they are either extraspecial of order p^3 for p=3,5, or elementary abelian, using the classification of finite simple groups.

Contribution

It proves a classification result for defect groups of blocks with all nontrivial subsections conjugate, leveraging the classification of finite simple groups.

Findings

Defect groups are either extraspecial of order p^3 for p=3,5, or elementary abelian.

All nontrivial subsections of the block are conjugate.

The classification relies on the finite simple groups.

Abstract

Suppose that all nontrivial subsections of a $p$-block $B$ are conjugate (where $p$ is a prime). By using the classification of the finite simple groups, we prove that the defect groups of $B$ are either extraspecial of order $p^3$ with $p \in \{3,5\}$ or elementary abelian.