The Inuence Of F~*-Subgroups On The Structure Of Finite Groups

Let F be a class of groups. A subgroup H of a group G is called the F*-subgroup of G if there exists a normal subgroup B of G such that HK(?) G, B/(B∩HG)∈F and B contains a Sylow q-subgroup of G for any prime q with (q,|H|)= 1.In this paper, by using some special subgroups (such as minimal subgroups, maximal subgroups, maximal subgroups and 2-maximal subgroups of Sylow subgroups) of G, we investigate the influence of F*-subgroups of G on supersolubility, p-nilpotent, solubility of a finite group. We obtain some sufficient conditions for a finite group to be solvable, supersoluble, nilpotent.The thesis is divided into four parts according to contents.In part 1, we give the new concept of.F*-subgroup, and introduce its investigative background, some definitions and previously known results. The relationship among the F*-subgroups,.F-s-subgroups,F-supplemented subgroups, and the conditional c-normal subgroups were studied. We give some properties about F*-subgroups and correlative lem-mas.In part 2, by using F*-subgroups of prime power order of G, some sufficient conditions of a finite group G to be supersoluble were obtained. We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group and every maximal subgroup of every Sylow subgroup of G is a U*-subgroup of G, Then G∈U.Theorem 2.1.2 Let G be a finite group. If H is a normal subgroup of G such that G/H∈U and every maximal subgroup of every Sylow subgroup of H is a U*-subgroup of G. Then G∈U.Theorem 2.1.3 Let G be a finite group. If G = AB, where B is a Hall-subgroup of G such that all Sylow subgroups of B are cyclic, A is subnormal in G, and every maximal subgroup of every Sylow subgroup of A is a U*-subgroup of G. Then G∈U. Theorem 2.1.4 Let G be a finite group. If H is a normal subgroup of G such that G/H∈U and every minimal subgroup of H is a W*-subgroup of G. Then G∈U.In part 3, by using maximal subgroups and 2-maximal subgroups of Sylow subgroups, we investigate the influence of F*-subgroups on p-nilpotency of finite groups. We obtain some main results as follows:Theorem 2.2.2 Let G be a finite group and p a prime divisor of |G| with (|G|, p-1)= 1. Then G is p-nilpotent if and only if G has a normal subgroup N such that G/N is p-nilpotent and every maximal subgroup of every Sylow subgroup of N is an Np*-subgroup of G.Theorem 2.2.4 Let G be a finite group and p a prime divisor of |G|with (|G|, p-1)= 1. Then G is p-nilpotent if and only if G has a normal subgroup N such that G/N is p-nilpotent and every 2-maximal subgroup of every Sylow subgroup of N is an Np*-subgroup of G.In part 4, by using F*-subgroups, some sufficient conditions of a finite group G to be soluble were obtained. We obtain some main results as follows:Theorem 2.3.1 Let G be a finite group. Then G is soluble if and only if there exists a p-subgroup of G to be an Sp*-subgroup of G.Theorem 2.3.2 Let K≤G and |G:K|be prime power. If every Sylow subgroup of K is an S*-subgroup of G, then G is soluble.