子群的半覆盖—避开性与有限群的结构

设G是有限群,H为G的子群.如果存在一个主群列1=G_0(?)G_1(?)…(?) G_(n-1)(?)G_n=G,使得对每个i=1,2,…,n,或者H覆盖G_i/G_(i-1),或者H避开G_i/G_(i-1),则称H为G的半覆盖—避开子群.利用G的Sylow子群的极大子群,Sylow子群的2-极大子群的半覆盖—避开性得到了群的可解性,p-幂零性的判别,同时得到了群系的一些结论.

Semi $CAP$-Subgroups and the Structure of Finite Groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a semi $CAP$- subgroup of $G$ if there exists a given chief series $1=G_0\lhd G_1\lhd \cdots \lhd G_{n-1}\lhd G_n=G$ such that for every $i=1, 2, \ldots, n$, either $H$ covers $G_i/G_{i-1}$ or $H$ avoids $G_i/G_{i-1}$. In this paper, using the condition that some maximal subgroups or $2$-maximal subgroups are semi $CAP$-subgroups of $G$, we get some criterions for solvability, and $p$-nilpotency of $G$. Meanwhile, we obtain some results about formation.