Freyd's generating hypothesis for groups with periodic cohomology

TL;DR

This paper investigates Freyd's generating hypothesis within the stable module category for finite groups with periodic cohomology, establishing precise conditions under which the hypothesis holds based on the Sylow p-subgroup structure.

Contribution

It characterizes when Freyd's generating hypothesis holds for groups with periodic cohomology, linking it to Sylow p-subgroups being cyclic of order 2 or 3.

Findings

GH holds iff Sylow p-subgroup is C2 or C3

Provides equivalent conditions for GH in periodic cohomology groups

Establishes a clear criterion for GH validity in this context

Abstract

Let $G$ be a finite group and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.