有限群的弱m-正规子群

On weak m-normal subgroups in finite group

定义了有限群G的弱m-正规子群,并在此定义下,赋予有限群的子群的诸多性质,得出(1)若G的Sylow-子群在G中弱m-正规,且至少有一个Sylow-子群在G的极大子群M中正规,则M为可解群.(2)若有限群G的Sylow-子群都是弱m-正规的,且在G的极大子群M中没有正规的Sylow-子群,则M是非可解的.(3)设有限群G的Sylow-子群都是弱m-正规的,G的极大子群M可解的充分必要条件是至少有一个Sylow-子群在M中正规.(4)若G的Sylow-子群都在G中弱m-正规,且至少有一个Sylow-子群在G的极大子群M中正规,M至少有3个不同的素因子,则G可解.(5)设M为G的任一极大子群,且M为可解群.若M的每个Sylow-子群非循环且它们的极大子群都在G中弱m-正规,则G可解.

With the concept of weak m-normal en ： （ 1 ） When all Sylow subgroups about G are normal subgroup of M, M is a maximal subgro subgroups, some results about weak m-normal subgroups are giv-weak m-normal subgroups, and there is a Sylow subgroup which is up of G, then M is a solvable group. （2） When all Sylow subgroups about G are weak m-normal subgroups, and there is no any Sylow subgroup which is normal subgroup of M , M is a maximal subgroup of G , then M is not a solvable group. （3） Let G be a finite group, and M is a maximal subgroup of G , all Sylow subgroups about G are weak m-normal subgroups, then M is a solvable group if and only if there is a Sylow subgroup which is normal subgroup of M. （4） When all Sylow subgroups about G are weak m-normal subgroups, and there is a Sylow subgroup which is normal subgroup of of M,M is a maximal subgroup ofG ,and ｜v｜=P1^a1P2^a2…Pn^an （n≥3） ,then finite groupG is a solvable group. （5）When M is a maximal subgroup of G, and M is a solvable group, all Sylow subgroups about M are not cyclic groups, and they are weak mnormal subgroups of G, then finite group G is a solvable group .