循环群的t-幂等自同态的个数

On the Numbers of t-Idempotent Endomorphisms of Cyclic Groups

设G为群,φ:G→G为群自同态。如果φ^(t)=φ,t∈N^(*),t≥2,那么称φ为G的t-幂等自同态。从新的角度认识循环群,并用初等数论的方法解决循环群幂等自同态问题。证明了若G是无限循环群,则G上t-幂等自同态的个数为2或3;若G为m阶循环群,m≥2,则G的2-幂等和3-幂等自同态的个数都可用m的不同的素因子的个数表示出来。

Let G be a group,φ:G→G a group endomorphism.Ifφ^(t)=φ,t∈N^(*),t≥2,thenφis called a t-idempotent endomorphism of G.This paper studies cyclic groups from a new perspective and solves the problem of idempotent endomorphisms of cyclic groups by means of elementary number theory.If G is an infinite cyclic group,then the number of t-idempotent endomorphisms on G is showed to be 2 or 3,and if G is a cyclic group of order m,m≥2,then the numbers of 2-idempotent and 3-idempotent endomorphisms of G are expressed by the number of different prime factors of m.