有限Abel群的一个特征(英文)

A CHARACTERISTIC OF FINITE ABELIAN GROUPS

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ，ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１，ａ~ｂ＝ａ^（１＋２）^（ｎ－１）＞同构，其中ｎ≥３，ｎ≥ｍ．利用该结果，作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群。

In this paper,it is shown that a finite group G is abelian if and only if G satisfies the following conditions : (Ⅰ) G has a power automorphism Q such that C_G(a) is an elementary abelian 2-group. (Ⅱ )G has no subgroup isomorphic to 2-group < a,b|a^2~n= b^2~m = 1,a^b = a^1+2^(n+1) > where n ≥ 3 and n≥m. In addition, applying the above result, the author also proves that if a finite group G has a power automorphism Q such that C_G(a) is an elementary abelian 2-group, then G is nilpotent.