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We let the reader prove that the identity is the coset N = N 1 and that the inverse of N a is N(a- 1 ). This group is denoted by GIN, and the definition of index gives IGINI = [G: N]. 22. , the function v: G -+ GIN = Na) is a surjective homomorphism with kernel N. Proof. The equation v(a)v(b) = v(ab) is just the formula NaNb = Nab; hence, v is a homomorphism. If Na E GIN, then Na = v(a), and so v is surjective. 8, so that N = ker v. • We have now shown that every normal subgroup is the kernel of some homomorphism.

H homomorphism if, for all a, bEG, f(a * b) = f(a) 0 IS a f(b). An isomorphism is a homomorphism that is also a bijection. We say that Gis isomorphic to H, denoted by G ~ H, if there exists an isomorphism f: G ..... H. The two-element groups G and H, whose multiplication tables are given above, are isomorphic: define f: G ..... H by f(l) = [0] and f( -1) = [1]. 3 This definition also applies to semigroups. Homomorphisms 17 Let f: G -+ H be an isomorphism, and let a 1 , a 2 , ••• , a" be a list, with no repetitions, of all the elements of G.

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