具有某些X-半置换子群的有限群(英文)

Finite Groups with Some X-semipermutable Subgroups

最近几年,利用子群的置换性质刻画有限群结构成为了人们感兴趣的课题.文献(J.Algebra,2007,315:31-41.)引入了X-半置换子群的概念:设X是有限群G的一个非空子集.G的一个子群A称为在G中X-半置换,如果A在G中有一个补T使得对于T的任意子群T1,存在x∈X使得ATx1=Tx1A.结合文献(Commun.Algebra,2008,36(6):2333-2340.)引入的p-群的特殊极大子群,利用这些极大子群的X-半置换性,通过对X的巧妙选择,得到有限群成为p-可解、p-幂零和超可解的若干充分条件,推广了若干熟知的相关结果.这些新结果将丰富和促进有限群结构的相关研究.

In recent years, it has been of interest to use some permutable properties of subgroups to characterize the structure of finite groups. Guo et al （J. Algebra ,2007,315：31-41. ） introdueed the eoncept of X-semipermutable subgroups ： Let X be a non-empty subset of a group G. A subgroup A of G is said to be X-semipermutable in G if A has a supplement T in G such that for every subgroup Tl of T there exists an element x E X such that ATx1=Tx1A. In this paper, by combining the special maximal subgroups of p-groups introduced by Li et al （ Commun. Algebra,2008,36 （6） ：2333-2340. ） and cleverly ehoieing X, we obtain some new sufficient conditions for a finite group to be p-supersolvable, p-nilpotent and supersolvable with X-semipermutation of the special maximal subgroups of Sylow subgroups. Our results generalize some known related results, which will enrich and promote the related study of structure of finite groups.