Abstract Algebra Theory and Applications - Computer Science ...

8.1 DEFINITION AND EXAMPLES 141To show that φ 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 a m ≠ a n .Let us suppose the contrary; that is, a m = a n . In this case a m−n = e, wherem − n > 0, which contradicts the fact that a has infinite order. Our mapis onto since any element in G can be written as a n for some integer n andφ(n) = a n .□Theorem 8.3 If G is a cyclic group of order n, then G is isomorphic to Z n .Proof. Let G be a cyclic group of order n generated by a and define amap φ : Z n → G by φ : k ↦→ a k , where 0 ≤ k < n. The proof that φ is anisomorphism is one of the end-of-chapter exercises.□Corollary 8.4 If G is a group of order p, where p is a prime number, thenG is isomorphic to Z p .Proof. The proof is a direct result of Corollary 5.7.The main goal in group theory is to classify all groups; however, it makessense to consider two groups to be the same if they are isomorphic. We statethis result in the following theorem, whose proof is left as an exercise.Theorem 8.5 The isomorphism of groups determines an equivalence relationon the class of all groups.Hence, we can modify our goal of classifying all groups to classifying allgroups up to isomorphism; that is, we will consider two groups to be thesame if they are isomorphic.Cayley’s TheoremCayley proved that if G is a group, it is isomorphic to a group of permutationson some set; hence, every group is a permutation group. Cayley’sTheorem is what we call a representation theorem. The aim of representationtheory is to find an isomorphism of some group G that we wish tostudy into a group that we know a great deal about, such as a group ofpermutations or matrices.Example 6. Consider the group Z 3 . The Cayley table for Z 3 is as follows.+ 0 1 20 0 1 21 1 2 02 2 0 1□