On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910854142754816 |
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| author | Singh, Mandeep Sharma, Mahak |
| author_facet | Singh, Mandeep Sharma, Mahak |
| contents | Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that the conjecture is true when a finite non-abelian $p$-group $G$ has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes $Φ(G)$ elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order $p$ which fixes $Φ(G)$ elementwise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor Singh, Mandeep Sharma, Mahak Group Theory Primary: 20D15, Secondary: 20D45 Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that the conjecture is true when a finite non-abelian $p$-group $G$ has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes $Φ(G)$ elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order $p$ which fixes $Φ(G)$ elementwise. |
| title | On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor |
| topic | Group Theory Primary: 20D15, Secondary: 20D45 |
| url | https://arxiv.org/abs/2503.00954 |