有限n-正规化子群

Finite n-Normalizer Groups

基于Ashrafi的想法,定义了n-正规化子群并对其进行研究。首先由定义得到n-正规化子群的一些基本性质。其次,对于任意的正整数n证明了n-正规化子群的存在性。再次,证明了对于有限群G,若#Norm(G)≤3,则G为幂零群;若假定|G|为奇数,则当#Norm(G)≤4时G为幂零群。最后,证明了若#Norm(G)=2,则G″=1;若#Norm(G)=3且G有交换的Sylow2-子群,则G=1。

Based on Ashrafi＇s idea, n-normalizer groups are defined and investigated. First, some elementary properties about n-normalizer groups are given. Secondly ,the existence of finite n-normalizer groups for every positive integer n are proved. Thirdly, the nilpotency and derived lengths of 2,3-normalizer groups are investigated. In particular, it is shown that G＂ = 1 if # Norm （G） =2, and G＂= 1 if # Norm （G）=3 and G has abelian Sylow 2-subgroups.