超可解群的一些充分条件

Some Sufficient Conditions for Supersoluability of Groups

主要讨论了群G的Sylow子群及其他子群的弱拟正规性对群的影响,从而得到原群G超可解的几个充分条件的定理:1)群G有指数为素数的可解正规子群H,若H的每个Sylow子群的极大子群在G中弱拟正规,则G超可解;2)群G有指数为素数的正规子群H,若H的Sylow子群及Sylow子群的2-极大子群皆在G内弱拟正规,则G超可解;3)设G=AB,A超可解,B是P-群,p=maxπ(G),若B与A的极大子群可交换且A弱拟正规于G,则G超可解;4)M为G的幂零极大子群,若M及其极大子群皆在G中弱拟正规,则G超可解。

In this paper we discuss the influence on an original finite group G when its Sylow subgroups and other subgroups are weakly quasi-normal, then we obtain some sufficient conditions for supersoluability of group G. 1 ） Suppose that H is a solvable normal dubgroup of G with its exponent being prime, if every Sylow maximal subgroup is weakly quasi-normal in G, then G is supersolvable. 2） Suppose that H is a solvable normal dubgroup of G with its exponent being prime, if every Sylow maximal subgroup and two-maximal subgroup are weakly quasi-normal in G, then G is supersolvable. 3 ） Suppose that G = AB, A is supersolvable and B is P- subgroup, p = max or（G）. If the maximal subgroups of A and B change each other, and A is weakly quasi-normal in G, then G is supersolvable. 4） Suppose that M is a nilpotent maximal subgroup of G, if M and its maximal subgroup are weakly quasi-normal in G. Then G is supersolvable.