有限群的弱s-置换子群

Weakly s-permutable subgroups of finite groups

如果对群G的任意Sy low子群T,存在元素x∈G,使H Tx=TxH,则群G的子群H称为在G中弱s-置换.利用子群的弱s-置换性得出下列结果:1)设F是包含超可解群系U的饱和群系,H为G的可解正规子群.如果G/H∈F,且H的任一Sy low子群的极大子群在G中弱s-置换,则G∈F.2)设F是包含超可解群系U的饱和群系,H为G的可解正规子群.如果G/H∈F,且F(H)的任一Sy low子群的极大子群在G中弱s-置换,则G∈F.

A subgroup H of a group G is called weakly s-permutable in G if for every Sylow subgroup T of G, there exists an element x∈G such that HT^x = T^xH. In this paper, it gets the following results by using the weakly s-permutabilities of subgroups of finite groups. 1）Let F be a saturated formation containing U, H a soluble normal subgroup of G. If G/H ∈ F and every maximal subgroup of any Sylow subgroup of H is weakly s-permutable in G, then G ∈ F. 2） Let F be a saturated formation containing U, H a soluble normal subgroup of G.If G/H ∈ F and every maximal subgroup of any Sylow subgroup of F（H） is weakly s-permutable in G, then G ∈ F.