S-θ-Completions Of The Maximal Subgroups And The Solvability Of Finite Groups

In this paper, we mainly study the solvability and theπ- solvability of the finite groupsby using the S-θ-completions of the Maximal Subgroups. There are two sections in ourresuts.In the first section, we study theπ-solvability of the finite groups under some conditionsby using the S-θ-completions of the finite groups. We obtain some su?cient and necesseryconditions forπ-solvability of a finite group.Some main results as follows:Theorem 2.1.1 Let G be a group. Then the group G isπ-solvable if and only if(1) for every maximal subgroup of G ,there exists a S-θ-completion C of M such thatC/CoreG(M) isπ-solvable.(2)If the indice of M in G is composite,then C￠G.Theorem 2.1.3 Let G be a p-solvable group. Then the group G isπ-solvable ifand only if for every M∈δp(G), there exists a normal S-θ-completion C of M such thatC/CoreG(M) isπ-solvable.Theorem 2.1.5 Let G be a p-solvable group. Then the group G isπ-solvable if andonly if for any two distinct maximal subgroups M1 and M2 of G inδp(G), whenever M1and M2 have a common S-θ-completion C such that C/CoreG(M1) and C/CoreG(M2) areπ-solvable.Theorem 2.1.7 Let G be a p-solvable group. Then the group G isπ-solvable if andonly if for any two distinct maximal subgroups M1 and M2 of G in p(G), whenever M1 andM2 have a common S-θ-completion C such that |G : M1|π= |G : M2|π= |C :CoreG(M1)|π=|C :CoreG(M2)|π.In the second section,we study solvability of the finite groups. We obtain some su?cientconditions for solvability of a finite group, which improve the resuts of some references. Somemain results as follows: Theorem 2.2.1 Let G be a p-solvable group. Then the group G is solvable if andonly if for every maximal subgroup M with composite indice in G, there exists a normalS-θ-completion C of M such that C/CoreG(M) is solvable.Theorem 2.2.2 Let G be a p-solvable group. Then the group G is solvable if andonly if for every maximal subgroup M with composite indice in G, there exists a subnormalS-θ-completion C of M such that C/CoreG(M) is nilpotent.Theorem 2.2.3 Let G be a p-solvable group. For every maximal subgroup M withcomposite indice in G, if there exists a S-θ-completion C of M such that C/CoreG(M) issupersolvable and |C/CoreG(M)|2≤2, then G is solvable.