| Finite Groups with Some Self-Conjugate-Permutable Subgroups |
| Received:April 08, 2023 Revised:January 06, 2024 |
| Key Words:
$SC$-group $PSC$-group self-conjugate-permutable subgroup maximal subgroup
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12061030) and the Natural Science Foundation of Hainan Province (Grant No.122RC652). |
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| Abstract: |
| A subgroup $H$ of a group $G$ is said to be self-conjugate-permutable if $HH^{x}=H^{x}H$ implies $H^{x}=H$ for any $x$ of $G$. A finite group $G$ is called an $SC$-group ($PSC$-group, respectively) if all cyclic subgroups of $G$ of order $2$ or order $4$ (prime order or order $4$, respectively) are self-conjugate-permutable in $G$. In this paper, we first investigate the structure of finite non-solvable groups all of whose second maximal subgroups are $SC$-groups; then we mainly investigate the structure of finite groups in which all of maximal subgroups of even order are $PSC$-groups. In fact, we describe the structure of finite groups which are not $PSC$-groups but all of whose maximal subgroups of even order are $PSC$-groups. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2024.02.006 |
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