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

TL;DR

The paper advances the understanding of endotrivial $kG$-modules by leveraging Grodal’s homotopy-theoretic perspective to compute the fundamental group of the orbit category on nontrivial $p$-subgroups for broad group classes. It proves a general formula for $\pi_1(\mathscr O^*_p(G))$ under $\,\Omega_1(S)\le Z(S)$ and fusion control by $N_G(S)$, yielding $K(G)\cong\operatorname{Hom}(N_G(S)/J,k^{\times})$ and, consequently, a description of $T(G)$ that extends Carlson–Thevenaz to groups with abelian Sylow $p$-subgroups. The authors then specialize to odd primes with split metacyclic Sylow $p$-subgroups, providing an explicit two-case description of $\pi_1(\mathscr O^*_p(G))$ depending on whether $E\subseteq Z(S)$, and, in the noncentral case, a more intricate presentation in terms of $N_G(S)$, $O^{p'}$-subgroups, and centralizers. This work not only generalizes known results for abelian Sylow $p$-subgroups but also completes a detailed treatment of the metacyclic scenario, enhancing the computation of endotrivial groups in a wider suite of finite groups.

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 $\operatorname{End}_kM$ 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.

Theorems & Definitions (28)

• Theorem A
• Theorem B
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Definition 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4