Normalizer Quotients of Symmetric Groups and Inner Holomorphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909394679103488 |
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| author | Entin, Alexei Tsang, Cindy |
| author_facet | Entin, Alexei Tsang, Cindy |
| contents | We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\leq S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalizer Quotients of Symmetric Groups and Inner Holomorphs Entin, Alexei Tsang, Cindy Group Theory We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\leq S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest. |
| title | Normalizer Quotients of Symmetric Groups and Inner Holomorphs |
| topic | Group Theory |
| url | https://arxiv.org/abs/2408.07133 |