Representations of the double Burnside algebra and cohomology of the extraspecial p-group II

TL;DR

This paper investigates the structure of the mod p cohomology of certain summands in the stable splitting of the classifying space of an extraspecial p-group, extending understanding of its algebraic and topological properties.

Contribution

It provides new insights into the representations of the double Burnside algebra and the cohomology of extraspecial p-groups, specifically for the case p odd, and explores stable splittings of classifying spaces.

Findings

Identifies specific summands in the stable splitting of BE

Describes the structure of the mod p cohomology for these summands

Connects the algebraic representations with topological decompositions

Abstract

Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We consider the mod p cohomology of summands in the stable splitting of p-completed classifying space BE. Moreover, we consider the stable splitting for some finite groups with Sylow p-subgroup E.