摘要
利用Deskins在 195 9年所定义的有限群的极大子群的指数复合 ,得到关于群的可解性和超可解性一些新的刻划 .主要结果如下 :(1)假设F′G ={M :M为G的包含某Sylow子群正规化子的极大子群 ,且 ｜G∶M｜为合数 } ,则下列命题是等价的 :(i)G是可解的 ;(ii)对于每个M ∈F′G,存在一个极大完备C ,使得对于任意x ∈G ,Cx M ,并且C/K(C)幂零 .(iii)对于每个M ∈F′G,存在一个极大完备C ,使得C/K(C)或可换 ,或者满足G =CM ,且C/K(C)的阶无平方因子 .(2 )有限群G是超可解的当且仅当对于每个M ∈F′G,存在一个极大完备C ,使得G =CM且C/K(C)的阶无平方因子 .
By using the concept of index complex of a maximal subgroup introduced by Deskins in 1959, some new results on the solvability and supersolvability of a finite group are obtained. The main results are as follows: (1) Let F′ G={M∶M be any maximal subgroup of a finite group G, which contains some normalizer of a Sylow subgroup of G ,and ｜G∶M｜is composite},the following are equivalent: (i) G is solvable; (ii) for each M∈F′ G, there exites a maximal completion C such that for any x∈G, C xM, and C/K(C) is nilpotent; (ⅲ) for each M∈F′ G, there exites a maximal completion C such that C/K(C) is abelian or G=CM and C/K(C) is of square-free order. (2) G is supersolvable if and only if for each M∈F′ G, there exites a maximal completion C such that G=CM and C/K(C) is of square-free order.
出处
《北京建筑工程学院学报》
2004年第3期73-76,共4页
Journal of Beijing Institute of Civil Engineering and Architecture
关键词
极大子群
指数复合
极大完备
正规完备
maximal subgroup
index complex
maximal completion
normal completion
