有限可解群的一些充分条件

Some Sufficient Conditions of Finite Solvable Groups

利用子群的弱c-正规性得到了有限可解群的一些条件.首先,得到了有限可解群的一个充要条件,即有限群G是可解群当且仅当G的任二相邻子群A,B有A在B中弱c-正规.其次,得到了有限可解群的一些充分条件,若有限群G满足下列条件之一,则G是可解群;G的任一极大子群M的Sylow子群均在G中弱c-正规;G的任一极大子群M的极大子群均在G中弱c-正规;假设H是G的Hallπ-子群且2∈π,如果N_G(H)是可解群且在G中弱c-正规.

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 Hall π-subgroup of g and 2∈ π, if NG（H） is solvable and it is weakly c-normal in G.