Simple Modules for Groups with Abelian Sylow 2-Subgroups are Algebraic

TL;DR

This paper proves that for finite groups with abelian Sylow 2-subgroups, all simple modules over a field of characteristic 2 are algebraic, indicating a well-behaved tensor structure.

Contribution

It establishes that simple modules are algebraic for groups with abelian Sylow 2-subgroups, extending understanding of module algebraicity in this context.

Findings

All simple modules are algebraic for groups with abelian Sylow 2-subgroups.

Supports the conjecture that this property extends to all abelian 2-blocks.

Highlights the connection between group structure and module algebraicity.

Abstract

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having this property is equivalent to the tensor structure being 'nice' for that module. In this paper we prove that if G is a group with abelian Sylow 2-subgroups, and p=2, then all simple modules for G are algebraic. We include the conjecture that this result holds for all abelian 2-blocks.