On the orbit category on nontrivial p-subgroups and endotrivial modules

2025/01/20 by Mazza, Nadia
#20C33 #FOS: Mathematics #Group Theory (math.GR) #Primary: 20C20 #Representation Theory (math.RT) #secondary 20E15

paper · doi:10.48550/arxiv.2501.11415

Abstract

Let p be a prime, let G be a finite group of order divisible by p, and let k be a field of characteristic p. An endotrivial kG-module is a finitely generated kG-module M such that its endomorphism algebra End_kM decomposes as the direct sum of a one-dimensional trivial kG-module and a projective kG-module. In this article, we determine the fundamental group of the orbit category on nontrivial p-subgroups of G for a large class of finite groups, and use Grodal's approach to describe the group of endotrivial modules for such groups. Hence, we improve on the results about the group of endotrivial modules for finite groups with abelian Sylow p-subgroups obtained by Carlson and Thévenaz. With some additional analysis, we then determine the fundamental group of the orbit category on nontrivial p-subgroups of G and the group of endotrivial kG-modules in the case when G has a metacyclic Sylow p-subgroup for p odd.

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