Some Sufficient Conditions of Finite Solvable Groups

Using the weakly c-normality of subgroups,we obtain some conditions of a finite solvable group. Firstly,a sufficient and necessary condition of a finite solvable group is obtained:G is solvable if and only if A is weakly c-normal in B for any two neighbor subgroups A,B of G.Finally,some sufficient conditions of a finite solvable group are proved:Let G be a finite group,if G satisfies with one of the following conditions,then G is solvable:let M be an arbitrary maximal subgroup,all Sylow subgroups of M are weakly c-normal in G;let M be a arbitrary maximal subgroup,all maximal subgroup of M are weakly c-normal in G;let H be a Hallπ-subgroup of g and 2∈π,if N_G(H) is solvable and it is weakly c-normal in G.