Characterizing inner automorphisms and realizing outer automorphisms
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913342387388416 |
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| author | Sambale, Benjamin |
| author_facet | Sambale, Benjamin |
| contents | We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite group, whose outer automorphism group is isomorphic to G. The first theorem was proved by Pettet using a graph-theoretical construction of Heineken-Liebeck. A Lie-theoretical proof of the second theorem was sketched by Cornulier in a MathOverflow post. Our proofs are purely group-theoretical. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02992 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizing inner automorphisms and realizing outer automorphisms Sambale, Benjamin Group Theory We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite group, whose outer automorphism group is isomorphic to G. The first theorem was proved by Pettet using a graph-theoretical construction of Heineken-Liebeck. A Lie-theoretical proof of the second theorem was sketched by Cornulier in a MathOverflow post. Our proofs are purely group-theoretical. |
| title | Characterizing inner automorphisms and realizing outer automorphisms |
| topic | Group Theory |
| url | https://arxiv.org/abs/2405.02992 |