有限群超可解的一些充分条件

摘要

Let X be a non-empty subset of G.A subgroup H of a finite group G is said to be X-s-semipermutable in G if H has a supplement T in G such that H is X-permutable with any Sylow subgroup of T for some x ∈ X.Let P be a sylow p-subgroup of a finite group G,and d a powerof p such that 1 ≤ d ＜ I P |.We derive some theorems and corollaries that extend known results concerning S-semipermutable subgroups.We obtained in this paper that if H ∩ Op(G) is X-s-semipermutable in G for all normal subgroups H of G with | H | =d,where X is a soluble normal subgroup of G,then either G is p-supersoluble or else | P ∩ Op(G) I ＞ d.%G是有限群且X是一个非空集合.若子群H在G中有补充T,且对任取X中的元x,H与T的任意Sylow子群是X-置换的,子群H被称为是在G中X-s-半置换的.令d是一个小于P的阶的p-子群的阶.推广了S-半置换子群的一些结果,利用X-s-半置换子群的性质进一步研究有限群,给出有限群超可解的一些结论.即可得到:对任意的d阶正规子群H和G的可解正规子群X,若H ∩ Op(G)在G中X-s-半置换的,则G是p-超可解的或者是I P ∩ Op(G)I＞d.