无限亚循环群的自同构群

Automorphism Groups of Infinite Metacyclic Groups

研究了无限亚循环群的自同构,得到了它们的自同构群,并且证明了下面的定理：无限亚循环群G的自同构都是内自同构当且仅当G同构于下列群之一：（i）ZξZ^p）,其中p为奇素数,l为正整数,ξ把Z的生成元1变为ξ（1）=m时,p m,并且Aut Z（pl）=Z^pl＊=Z（pl-1）（p-1））=〈m〉;（ii）无限二面体群D∞.

We discuss automorphisms of infinite metacyclic groups and present their automorphism groups,and then we prove the following result：All automorphisms of an infinite metacyclic group G are inner automorphisms if and only if G is isomorphism to one of following groups：（i） Z ξZp^l,where p is an odd prime,l is an integer,ξmaps the generator 1 of Z toξ（1）=m,p m,and Aut Zp-l=Zp^l＊=Z（p^（1-1）（p-1））=〈m〉; （ii） The infinite dihedral group D∞.