The Influence Of Conjugate-permutable Subgroups And S-conditionally Permutable Subgroups On The Structure Of Finite Groups

The purpose of this paper is to study the influence of conjugate-permutable subgroupsand s-conditionally permutable subgroups on the structure of finite groups such as nilpotency,supersolvability, p-supersolvability. It is divided into two chapters. In the first chapter, weintroduce the investigative background of this paper and present some preliminary notions,properties and correlative lemmas, theorems. In the second chapter, we investigate thestructure of a group and obtain main results of this paper by using the conjugate-permutableproperty and s-conditionally permutable property of subgroups.Theorem 2.1.1 Let G be a group. If all subgroups of prime order of G are conjugate-permutable in G and for any subgroup H of G, Frattini subgroup of H is 1, then G isnilpotent.Theorem 2.1.2 Let G be a group of order odd. If all subgroups of prime order of Gare self-conjugate-permutable in G and for any subgroup H of G, Frattini subgroup of H is1, then G is nilpotent.Theorem 2.1.3 Let N be a normal subgroup of a group G. If G/N is nilpotent, everycyclic subgroup of N is conjugate-permutable in G and G is irrelevant to a p-elementaryinner-abelian subgroup of order prq, then G is nilpotent.Theorem 2.1.4 Let M be a nilpotent maximal subgroup of a group G and M G.If all subgroups of prime order of M are self-conjugate-permutable in G, all subgroups oforder 2 of G are conjugate-permutable in G and Frattini subgroup of each subgroup of G is1, then G is nilpotent.Theorem 2.1.5 Let G = HK. If all Sylow subgroups of subgroups H and K areconjugate-permutable in G, then G is nilpotent. Theorem 2.1.6 Let G = HK, P∈Sylp(G), P G and G′nilpotent. If all subgroupsof a subgroup H are conjugate-permutable in KP and all subgroups of a subgroup K areconjugate-permutable in HP, then G is nilpotent.Theorem 2.2.1 Let G be a group. Suppose every element of P~*(G) is s-conditionallypermutable in G, then G is supersolvable.Theorem 2.2.2 Let G be a group. G is supersolvable if and only if G has a normal sub-group H such that G/H is supersolvable and every element of P~*(F~*(H)) is s-conditionallypermutable in G.Theorem 2.2.3 Let F be a saturated formation containing U (the class of supersolv-able groups) and G a group. Then G∈F if and only if G has a normal subgroup H suchthat G/H∈F and every element of P?(F?(H)) is s-conditionally permutable in G.Theorem 2.3.1 Let G be a p-solvable group and p the smallest prime divisor of |G|. IfG is A4-free and every 2-maximal subgroup of any Sylow p-subgroup P of G is s-conditionallypermutable in G, then G is p-supersolvable.Theorem 2.3.2 Let G be a group and p the smallest prime divisor of |G|. If G isA4-free, then G is p-supersolvable if and only if G has a p-solvable normal subgroup N suchthat G/N is p-supersolvable and every 2-maximal subgroup of any Sylow p-subgroup P ofN is s-conditionally permutable in G.Theorem 2.3.3 Let G be a p-solvable group and p∈π(G) such that (|G|, p~2-1) = 1.If every 2-maximal subgroup of any Sylow p-subgroup P of G is s-conditionally permutablein G,then G is p-supersolvable.