p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且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U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-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.

A Characterization of p-Hypercyclically Embedded Subgroups of Finite Groups

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.