An upper bound for the representation dimension of group algebras with an elementary abelian Sylow $p$-subgroup

TL;DR

This paper establishes an upper bound for the representation dimension of group algebras with elementary abelian Sylow p-subgroups, using separable equivalence and Mackey decomposition techniques.

Contribution

It introduces a new upper bound for the representation dimension of such group algebras, linking group structure to algebraic complexity.

Findings

Representation dimension is at most |P| for groups with elementary abelian Sylow p-subgroups.

Uses separable equivalence to relate group algebras and their Sylow p-subgroups.

Employs Mackey decomposition to derive the upper bound.

Abstract

Linckelmann showed in 2011 that a group algebra is separably equivalent to the group algebra of its Sylow p-subgroups. In this article we use this relationship, together with Mackey decomposition, to demonstrate that a group algebra of a group with an elementary abelian Sylow $p$-subgroup $P$, has representation dimension at most $|P|$.