Endotrivial modules for cyclic $p$-groups and generalized quaternion groups via Galois descent

TL;DR

This paper offers new homotopical proofs for the classification of endotrivial modules over cyclic p-groups and quaternion groups, using Galois descent and spectral sequences, enhancing understanding of classical results.

Contribution

It introduces homotopical methods and Galois descent to compute endotrivial modules, providing a conceptual framework for classical classifications.

Findings

Homotopical proofs for cyclic p-groups and quaternion groups

Application of Galois descent and spectral sequences

Deeper conceptual understanding of classical results

Abstract

In this paper, we investigate the group of endotrivial modules for certain $p$-groups. Such groups were already been computed by Carlson-Th\'evenaz using the theory of support varieties; however, we provide novel homotopical proofs of their results for cyclic $p$-groups, the quaternion group of order 8, and for generalized quaternion groups using Galois descent and Picard spectral sequences, building on results of Mathew and Stojanoska. Our computations provide conceptual insights into the classical work of Carlson-Th\'evenaz.