Abstract Algebra Theory and Applications - Computer Science ...

8IsomorphismsMany groups may appear to be different at first glance, but can be shownto be the same by a simple renaming of the group elements. For example,Z 4 and the subgroup of the circle group T generated by i can be shownto be the same by demonstrating a one-to-one correspondence between theelements of the two groups and between the group operations. In such acase we say that the groups are isomorphic.8.1 Definition and ExamplesTwo groups (G, ·) and (H, ◦) are isomorphic if there exists a one-to-oneand onto map φ : G → H such that the group operation is preserved; that is,φ(a · b) = φ(a) ◦ φ(b)for all a and b in G. If G is isomorphic to H, we write G ∼ = H. The map φis called an isomorphism.Example 1. To show that Z 4∼ = 〈i〉, define a map φ : Z4 → 〈i〉 by φ(n) = i n .We must show that φ is bijective and preserves the group operation. Themap φ is one-to-one and onto becauseφ(0) = 1φ(1) = iφ(2) = −1φ(3) = −i.Sinceφ(m + n) = i m+n = i m i n = φ(m)φ(n),138