Abstract Algebra Theory and Applications - Computer Science ...

142 CHAPTER 8 ISOMORPHISMSThe addition table of Z 3 suggests that it is the same as the permutationgroup G = {(0), (012), (021)}. The isomorphism here is( ) 0 1 20 ↦→= (0)0 1 2( ) 0 1 21 ↦→= (012)1 2 0( ) 0 1 22 ↦→= (021).2 0 1Theorem 8.6 (Cayley) Every group is isomorphic to a group of permutations.Proof. Let G be a group. We must find a group of permutations G thatis isomorphic to G. For any g ∈ G, define a function λ g : G → G byλ g (a) = ga. We claim that λ g is a permutation of G. To show that λ g isone-to-one, suppose that λ g (a) = λ g (b). Thenga = λ g (a) = λ g (b) = gb.Hence, a = b. To show that λ g is onto, we must prove that for each a ∈ G,there is a b such that λ g (b) = a. Let b = g −1 a.Now we are ready to define our group G. LetG = {λ g : g ∈ G}.We must show that G is a group under composition of functions and findan isomorphism between G and G. We have closure under composition offunctions sinceAlso,and(λ g ◦ λ h )(a) = λ g (ha) = gha = λ gh (a).λ e (a) = ea = a(λ g −1 ◦ λ g )(a) = λ g −1(ga) = g −1 ga = a = λ e (a).We can define an isomorphism from G to G by φ : g ↦→ λ g . The groupoperation is preserved sinceφ(gh) = λ gh = λ g λ h = φ(g)φ(h).