Some Researches On Generalized Permutability Of Subgroups

A subgroup H of a group G is said to be weakly s-permutablein G if there is a subnormal subgroup K of G such that G = HK and H∩K≤HsG,where HsG is the subgroup generated by all s-permutable subgroup of G contained in H.In this paper,p-supersolvability and p-nilpotency of finite group are studied by investigating on the weakly s-permutability of some subgroup.Also,anopen question of Skiba areattackedand some new results are obtained.We obtain mainly the following theorems:Theorem 3.2.1 Let G be a p-solvable group.If for any non-Frattini p-chief factor H/K of G,there is a maximal subgroup P1 of some Sylow p-subgroup of G such that P1 does not cover H/K and is weakly s-supplemented in G,then G is p-supersolvable.Theorem 3.3.1Let be a prime divisor of group G and(p-1,|G|)= 1.If for any non-Frattini p-chief factor H/K of G.there is a maximal subgroup P1 of some Sylow p-subgroup of G such that P1 does not cover H/K and is weakly s-supplemented in G,then G is p-supersolvable.Theorem 4.2,1 Let(?)be a saturated formation containing all supersolvable groups and G a group with a normal subgroups E such that G/E ∈(?).Suppose that every non-cyclic Sylow subgroup P of F*(E)has a subgroup D such that 1<|D|<|P| and all subgroups H of P with order |H|=|D| and with order 2|D|(if P is a non-abelian 2-group and |P:D|>2)having no supersolvable supplement in G are weakly s-supplemented inG.If one of the following holds:(ⅰ)P is impartible normal in G;(ⅱ)P ∩ Φ(G)≠P’;(ⅲ)|D|≤|P’|;(ⅳ)|D|≥|P’| and(l(P/P’),l(|D|/|P’|)=1 or(l(P),ll(|P:D|))=1 Then G ∈(?).