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

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.