如何判断两个系统的同构？ Theorem 9.7.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 Z-G by o:n a".Then o(m+n)amtn a"a"o(m)o(n). 观察 To show that o is injective,suppose that m and n are two elements in Z, where m 4n.We can assume that m >n.We must show that am 4 a". 构造 Let us suppose the contrary;that is,am=a".In this case am-n=e,where m-n >0,which contradicts the fact that a has infinite order.Our map 证明（定义) 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（定义） 如何判断两个系统的同构？