关于极小子群的中心化子

On the Centralizers of the Minimal Subgroups of Finite Groups

子群的中心化子对群的结构有很强的控制作用 .称有限群G为PNC群 ,如果G的每个极小子群X均满足CG(X) =NG(X) .首先证明了PNC群是介于幂零群与 2 闭群之间的一类可解群 .其次 ,考虑极小子群的中心化子与群的可解性的关系 。

It's well known that the properties of the centralizers of subgroups of a finite group have a great influence on the structure of the group. If, for any minimal subgroup X of G, C G(X)=N G(X), G is called a PNC group. Firstly, we prove that PNC groups are solvable groups whose properties are between nilpotent groups and 2-closed groups. Secondly, we investigate the relationship between the centralizers of minimal subgroups and the solvability of a finite group. Some new sufficient conditions about the solvability of finite groups are obtained.