Spherical $p$-group complexes arising from finite groups of Lie type
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917292111036416 |
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| author | Piterman, Kevin Iván |
| author_facet | Piterman, Kevin Iván |
| contents | We show that the $p$-group complex of a finite group $G$ is homotopy equivalent to a wedge of spheres of dimension at most $n$ if $G$ contains a self-centralising normal subgroup $H$ which is isomorphic to a group of Lie type and Lie rank $n$ in characteristic $p$. If in addition every order-$p$ element of $G$ induces an inner or field automorphism on $H$, the $p$-group complex of $G$ is $G$-homotopy equivalent to a spherical complex obtained from the Tits building of $H$.
We also prove that the reduced Euler characteristic of the $p$-group complex of a finite group $G$ is non-zero if $G$ has trivial $p$-core and $H$ is a self-centralising normal subgroup of $G$ which is a group of Lie type (in any characteristic), except possibly when $p=2$ and $H=A_n(4^a)$ ($n\geq 2$) or $E_6(4^a)$. In particular, we conclude that the Euler characteristic of the $p$-group complex of an almost simple group does not vanish for $p\geq 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spherical $p$-group complexes arising from finite groups of Lie type Piterman, Kevin Iván Group Theory Algebraic Topology 05E18, 06A11, 20D30, 20E42, 20G40, 55U05 We show that the $p$-group complex of a finite group $G$ is homotopy equivalent to a wedge of spheres of dimension at most $n$ if $G$ contains a self-centralising normal subgroup $H$ which is isomorphic to a group of Lie type and Lie rank $n$ in characteristic $p$. If in addition every order-$p$ element of $G$ induces an inner or field automorphism on $H$, the $p$-group complex of $G$ is $G$-homotopy equivalent to a spherical complex obtained from the Tits building of $H$. We also prove that the reduced Euler characteristic of the $p$-group complex of a finite group $G$ is non-zero if $G$ has trivial $p$-core and $H$ is a self-centralising normal subgroup of $G$ which is a group of Lie type (in any characteristic), except possibly when $p=2$ and $H=A_n(4^a)$ ($n\geq 2$) or $E_6(4^a)$. In particular, we conclude that the Euler characteristic of the $p$-group complex of an almost simple group does not vanish for $p\geq 7$. |
| title | Spherical $p$-group complexes arising from finite groups of Lie type |
| topic | Group Theory Algebraic Topology 05E18, 06A11, 20D30, 20E42, 20G40, 55U05 |
| url | https://arxiv.org/abs/2403.07489 |