关于极大子群的指数复合

On the Index Complex of a Maximal Subgroup

利用Deskins在 195 9年所定义的有限群的极大子群的指数复合 ,给出了有限群为π -可解 ,可解 ,超可解 。

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 xM, 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.