| 邵长国,蒋琴会.素数幂、双素数幂阶元的共轭类长的个数为4的 有限群的结构[J].数学年刊A辑,2018,39(1):1~6 |
| 素数幂、双素数幂阶元的共轭类长的个数为4的 有限群的结构 |
| Structure of Finite Groups with Four Conjugacy Class Sizes of Primary and Biprimary Elements |
| Received:January 12, 2016 Revised:June 13, 2016 |
| DOI：10.16205/j.cnki.cama.2018.0001 |
| 中文关键词: 有限群, 素数幂、双素幂阶元, 共轭类长, 2-Frobenius群 |
| 英文关键词:Finite groups, Primary and biprimary elements, Conjugacy class sizes, 2-Frobenius groups |
| 基金项目:本文受到国家自然科学基金 (No.11301218)和山东省自然科学基金项目 (No.ZR2014AM020) 的资助. |
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| 中文摘要: |
| 设群$G$为一个有限群. 如果群$G$中素数幂、双素幂阶元的共轭类长的集合为$\{1,p^a,m,p^bm\}$,
那么群$G$是可解的, 其中$a>b$为正整数, $p$为素数且与$m$互素. 进一步,
给出了群$G/\textbf{Z}(G)$的结构, 这是对文``Chen R F, Zhao X H. A criterion for a group to have nilpotent
$p$-complements [J]. {\it Monatsh Math}, 2016, 179(2):221--225''中定理A主要结论的一个推广. |
| 英文摘要: |
| Let $G$ be a finite group. It is proved that $G$ is solvable if the set of its
conjugacy class sizes of primary and biprimary elements is $\{1, p^a, m, p^bm\}$,
where $a > b$ are two positive integers and $p$ is a prime coprime to integer
$m$. Moreover, the authors give a detailed structure description of $G/\textbf{Z}(G)$,
which generalizes the main result of Theorem A in ``Chen, R. F. and Zhao, X. H., {\it Monatsh. Math.}, 2016, 179(2):221--225''. |
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