有限n-正规化子群

北京大学学报（自然科学版）

有限n-正规化子群

刘燕俊¹,朱一心²

1.北京大学数学科学学院，北京100871；2.首都师范大学数学科学学院，北京100037;

收稿日期:2008-01-17 出版日期:2009-01-20 发布日期:2009-01-20

Finite n-Normalizer Groups

LIU Yanjun¹，ZHU Yixin²

1. School of Mathematical Sciences, Peking University, Beijing 100871; 2. School of Mathematical Sciences, Capital Normal University, Beijing 100037;

Received:2008-01-17 Online:2009-01-20 Published:2009-01-20

摘要/Abstract

摘要： 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。

关键词: 有限群, n-正规化子群, 幂零群, 导列长

Abstract: 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.

Key words: finite groups, n-normalizer groups, nilpotency, derived length

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