The Influence Of S-completions Of The Maximal Subgroups On The Structure Of Finite Groups

In the investigation of finite groups, using the maximal subgroups to characterize the structure of group G is a regular and effective method.In 1959,the concept of maximal com-pletion was introduced by Deskins in [1],and he obtain a sufficient and necessary condition for a finite group to be solvable.The question of this have achieved some great progress.for example, A.Ballester-Bolinches and Luis M.Ezquerro have achieved a sufficient and necessary condition for a finite group to be supersolvable:theorem 1.1.3.Shirong Li and Yaoqing Zhao have achieved some great progress,and have givin the following definition:s-completion. The purpose of this paper is to study the influence of s-completions of the maximal subgroups on the structure of finite groups such as solvability, supersolvability, nilpotency. The paper is divided into two chapters. In the first chapter, we introduce the investigative background, the preliminary notions, lemmas and the correlative theorems. In the second chapter, we use the properties of maximal completions and s-completions of the maximal subgroups to investigate the structure of finite groups, we obtain some sufficient and necessary conditions for a finite group to be solvable, to be supersolvable and nilpotent. Some main results as follows:Theorem 2.1.4 Let G be a finite group, where p is a prime divisor of|G|. Then G is solvable if and only if for each maximal subgroup such that|G:M|p= 1, there exists a maximal completion C such that C/K(C) is nilpotent with Sylow 2-group of class at most 2.Theorem 2.1.5 Let G be a finite group. If for each c-maximal subgroup M, there exists a maximal completion C such that|C/K(C)|is of square free order, and|C/K(C)|≥η(G:M). In addition,η(G:M)2≤4. Then G is solvable. In particular, G either is 2-supersolvable or has a homomorphic image isomorphic to the symmetric group S4.Theorem 2.1.6 Let G be a finite group, and G has one solvable maximal subgroup M of index prime power. If for every maximal subgroup M of composite index and r2(M)≤2, there exists a maximal completion C such that|C/K(C)|is of square free order. then G is either solvable or A5 is involved in G. Theorem 2.2.1 Let G be a finite solvable group. If for every pc-maximal subgroup M, there exists a maximal completion such that C/K(C) is cyclic of composite order and (|C/K(C)|, (pd-1)(pd-1-1)…(p-1))=1, then G either is supersolvable or has a homo-morphic image isomorphic to the symmetric group S4.Theorem 2.2.3 Let G be a finite group and S4-free, then G is supersolvable if and only if for each c-maximal subgroup, there exists a s-completion C such that CM=G and C/K(C) is cyclic.Theorem 2.2.5 Letπbe a set of primes and G be aπ-solvable finite group. Then G is 7r-supersolvable if and only if for each c-maximal subgroup M of G with|G:M|π≠1, there exists an s-completion C such that CM=G and C/K(C) is cyclic of order p, for some p∈π.