On the orbit category on nontrivial $p$-subgroups and endotrivial modules
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917898286530560 |
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| author | Mazza, Nadia |
| author_facet | Mazza, Nadia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the orbit category on nontrivial $p$-subgroups and endotrivial modules Mazza, Nadia Representation Theory Group Theory Primary: 20C20, 20C33, secondary 20E15 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. |
| title | On the orbit category on nontrivial $p$-subgroups and endotrivial modules |
| topic | Representation Theory Group Theory Primary: 20C20, 20C33, secondary 20E15 |
| url | https://arxiv.org/abs/2501.11415 |