Section 1.5. Homomorphisms 41 Example 1.61. (i) If μ2 is the multiplicative group μ2 = {±1}, then sgn: Sn → μ2 is a homo- morphism, by Theorem 1.13. The kernel of sgn is the alternating group An, the set of all even permutations, and its image is μ2. (ii) Determinant is a surjective homomorphism det: GL(n, R) → R×, the mul- tiplicative group of nonzero real numbers, whose kernel is the special linear group SL(n, R) of all n×n matrices of determinant 1, and whose image is R× (det is surjective: if a ∈ R×, then det: [ a 0 0 1 ] → a). (iii) Let H = a be a cyclic group of order n, and define f : Z → H by f(k) = ak. Then f is a homomorphism with ker f = n . Proposition 1.62. Let f : G → H be a homomorphism. (i) ker f is a subgroup of G and im f is a subgroup of H. (ii) If x ∈ ker f and a ∈ G, then axa−1 ∈ ker f. (iii) f is an injection if and only if ker f = {1}. Proof. (i) Routine. (ii) f(axa−1) = f(a)1f(a)−1 = 1. (iii) f(a) = f(b) if and only if f(b−1a) = 1. • Definition. A subgroup K of a group G is called a normal subgroup if k ∈ K and g ∈ G imply gkg−1 ∈ K. If K is a normal subgroup of G, we write K G. Proposition 1.62(ii) says that the kernel of a homomorphism is always a normal subgroup (the converse is Corollary 1.75). If G is an abelian group, then every subgroup K is normal, for if k ∈ K and g ∈ G, then gkg−1 = kgg−1 = k ∈ K. The converse of this last statement is false: in Proposition 1.69, we shall see that there is a nonabelian group of order 8 (the quaternions), each of whose subgroups is normal. The cyclic subgroup H = (1 2) of S3, consisting of the two elements (1) and (1 2), is not a normal subgroup of S3: if α = (1 2 3), then α(1 2)α−1 = (1 2 3)(1 2)(3 2 1) = (2 3) / ∈ H [alternatively, Theorem 1.9 gives α(1 2)α−1 = (α1 α2) = (2 3)]. On the other hand, the cyclic subgroup K = (1 2 3) of S3 is a normal subgroup, as the reader should verify. It follows from Examples 1.61(i) and (ii) that An is a normal subgroup of Sn and SL(n, R) is a normal subgroup of GL(n, R) (it is also easy to prove these facts directly). Definition. Let G be a group. A conjugate of a ∈ G is an element in G of the form gag−1 for some g ∈ G.

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