Fixed point subgroups of a supertight automorphism
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929224164573184 |
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| author | Karhumäki, Ulla |
| author_facet | Karhumäki, Ulla |
| contents | Let $G$ be an infinite simple group of finite Morley rank and $α$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(α^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$. It is know (using CFSG) that the socle $S_n:={\rm Soc}(P_n)$ is a (twisted) Chevalley group over a pseudofinite field. We prove that there is $r\in \mathbb{N}\setminus\{0\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that there is no $m \in \mathbb{N}\setminus \{0\}$ so that for each $n$ the sizes of the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the recent identification result of $G$ under the assumption ${\rm pr}_2(G)=1$, the use of CFSG is not needed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed point subgroups of a supertight automorphism Karhumäki, Ulla Group Theory Logic Let $G$ be an infinite simple group of finite Morley rank and $α$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(α^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$. It is know (using CFSG) that the socle $S_n:={\rm Soc}(P_n)$ is a (twisted) Chevalley group over a pseudofinite field. We prove that there is $r\in \mathbb{N}\setminus\{0\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that there is no $m \in \mathbb{N}\setminus \{0\}$ so that for each $n$ the sizes of the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the recent identification result of $G$ under the assumption ${\rm pr}_2(G)=1$, the use of CFSG is not needed. |
| title | Fixed point subgroups of a supertight automorphism |
| topic | Group Theory Logic |
| url | https://arxiv.org/abs/2401.14222 |