On centers of blocks with one simple module

TL;DR

This paper characterizes the algebra structure of the center of a specific type of block in finite group theory, focusing on blocks with elementary abelian defect groups of order 16 and a single simple module.

Contribution

It provides a detailed description of the algebra structure of the center for non-nilpotent blocks with these properties, extending previous analyses in the field.

Findings

Describes the algebra structure of the center of such blocks.

Extends understanding of blocks with elementary abelian defect groups.

Builds on prior work related to 3-blocks of defect 2.

Abstract

Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main result of this paper describes the algebra structure of the center of B. This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012].