如何判断两个系统的同构？ Theorem 9.2 All cyclic groups of infinite order are isomorphic to Z. PRoOF.Let G be a cyclic group with infinite order and suppose that a is a generator of G.Define a map:ZG byo:n a".Then 观察 o(m+n)amtn a"a"o(m)o(n). 构造 To show that o is injective,suppose that m and n are two elements in Z, where m n.We can assume that m >n.We must show that am a". 证明 Let us suppose the contrary;that is,am=a".In this case am-=e,where m-n >0,which contradicts the fact that a has infinite order.Our map is onto since any element in G can be written as an for some integer n and o(n)a". ▣观察 构造 证明 如何判断两个系统的同构？