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

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.
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''.