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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2406.10623 |
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| _version_ | 1866916288729710592 |
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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 if $G$ is an odd order finite non-abelian monolithic $p$-group such that every maximal subgroup of $G$ is non-abelian and $[Z(M), g] \leq Z(G)$ for every maximal subgroup $M$ of $G$ and $g \in G \setminus M$. Then $G$ has a non-inner automorphism of order $p$ leaving the Frattini subgroup $Φ(G)$ elementwise fixed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the conjecture of non-inner automorphisms of finite $p$-groups Singh, Mandeep Sharma, Mahak Group Theory 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 if $G$ is an odd order finite non-abelian monolithic $p$-group such that every maximal subgroup of $G$ is non-abelian and $[Z(M), g] \leq Z(G)$ for every maximal subgroup $M$ of $G$ and $g \in G \setminus M$. Then $G$ has a non-inner automorphism of order $p$ leaving the Frattini subgroup $Φ(G)$ elementwise fixed. |
| title | On the conjecture of non-inner automorphisms of finite $p$-groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2406.10623 |