On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912415046696960 |
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| author | Meyer, Anja |
| author_facet | Meyer, Anja |
| contents | Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on $n>1$. Finally, we use the theory of fusion systems to describe the ring of stable elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04720 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ Meyer, Anja Algebraic Topology Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on $n>1$. Finally, we use the theory of fusion systems to describe the ring of stable elements. |
| title | On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2506.04720 |