p-超循环嵌入子群的一个判别准则
A Characterization of p-Hypercyclically Embedded Subgroups of Finite Groups

令$E$是有限群$G$的一个正规子群, 且$\mathcal{U}$是所有有限超可解群的集合. $E$称为在$G$中是$p${-}超循环嵌入的, 如果$E$的每个$pd${-}阶的$G${-}主因子是循环的. $G$的子群$H$称为在$G$中是$\mathcal{U}$-$\Phi${-}可补充的, 如果存在$G$的一个次正规 子群$T$, 使得$G=HT$, 且$(H\cap T)H_{G}/H_{G}\leq\Phi(H/H_{G})Z_{\mathcal{U}}(G/H_{G})$, 其中$Z_{\mathcal{U}}(G/H_{G})$是商群$G/H_{G}$的$\mathcal{U}${-}超中心. 作者证明, 如果$E$的一些$p${-}子群在$G$中是$\mathcal{U}$-$\Phi${-}可补充的, 那么$E$在$G$中是$p${-}超循环嵌入的. 作为应用, 得到了有限群是$p${-}超可解的若干判断准则, 并且推广了一些已知的结果.
Let $E$ be a normal subgroup of a finite group $G$ and $\mathcal{U}$ the class of all finite supersolvable groups. $E$ is said to be $p$-hypercyclically embedded in $G$ if every $pd$-$G$-chief factor below $E$ is cyclic. A subgroup $H$ of $G$ is $\mathcal{U}$-$\Phi$-supplemented in $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G=HT$ and $(H\cap T)H_{G}/H_{G}\leq\Phi(H/H_{G})Z_{\mathcal{U}}(G/H_{G})$, where $Z_{\mathcal{U}}(G/H_{G})$ is the $\mathcal{U}$-hypercentre of $G/H_{G}$. In this paper, it is proved that $E$ is $p$-hypercyclically embedded in $G$ if some classes of $p$-subgroups of $E$ are $\mathcal{U}$-$\Phi$-supplemented in $G$. As applications, some new characterizations of $p$-supersolvability of finite groups are obtained and some recent results are extended.