有限群的SS-半置换p-子群与p-幂零性

On SS-Semipermutable p-Subgroups and p-Nilpotency of Finite Groups

设G是有限群,H是群G的子群.如果群G中存在子群B使得G=HB,并且H与B的所有Sylow p-子群置换,其中素数p满足(p,|H|)=1,则称H在G中SS-半置换.假设P是群G的Sylow p-子群,D是P的非平凡子群.利用有限群G的Sylow p-子群P的|D|阶子群的SS-半置换性来研究有限群G的结构,给出了有限群G是p-幂零群的两个充分条件.

Let G is a finite group,and H is a subgroup of G.If there exists a subgroup B of G such that G=HB and H permutes with all of Sylow p-subgroup of B,where prime p is a coprime with the order of H,then H is SS-semipermutable group of G.Suppose P is a Sylow p-subgroup of group G and D is a nontrivial subgroup of P.In this note,two sufficient conditions for a Sylow p-nilpotency of finite group G are obtained by using the SS-semipermutation of some subgroups of Sylow p-subgroup of group G.