42 Chapter 1. Groups I It is clear that a subgroup K ⊆ G is a normal subgroup if and only if K contains all the conjugates of its elements: if k ∈ K, then gkg−1 ∈ K for all g ∈ G. Example 1.63. (i) Theorem 1.9 states that two permutations in Sn are conjugate if and only if they have the same cycle structure. (ii) In Linear Algebra, two matrices A, B ∈ GL(n, R) are called similar if they are conjugate that is, if there is a nonsingular matrix P with B = PAP −1. We shall see that A and B are conjugate if and only if they have the same rational canonical form (Theorem 8.36). Definition. If G is a group and g ∈ G, define conjugation γg : G → G by γg(a) = gag−1 for all a ∈ G. Proposition 1.64. (i) If G is a group and g ∈ G, then conjugation γg : G → G is an isomorphism. (ii) Conjugate elements have the same order. Proof. (i) If g, h ∈ G, then (γgγh)(a) = γg(hah−1) = g(hah−1)g−1 = (gh)a(gh)−1 = γgh(a) that is, γgγh = γgh. It follows that each γg is a bijection, for γgγg−1 = γ1 = 1 = γg−1 γg. We now show that γg is an isomorphism: if a, b ∈ G, γg(ab) = g(ab)g−1 = ga(g−1g)bg−1 = γg(a)γg(b). (ii) If a and b are conjugate, there is g ∈ G with b = gag−1 that is, b = γg(a). But γg is an isomorphism, and so Exercise 1.49 on page 44 shows that a and b = γg(a) have the same order. • Example 1.65. The center of a group G, denoted by Z(G), is Z(G) = {z ∈ G : zg = gz for all g ∈ G}. Thus, Z(G) consists of all elements commuting with everything in G. It is easy to see that Z(G) is a subgroup of G it is a normal subgroup because if z ∈ Z(G) and g ∈ G, then gzg−1 = zgg−1 = z ∈ Z(G). A group G is abelian if and only if Z(G) = G. At the other extreme are groups G with Z(G) = {1} such groups are called centerless. For example, Z(S3) = {(1)} indeed, all large symmetric groups are centerless, for Exercise 1.25 on page 16 shows that Z(Sn) = {(1)} for all n ≥ 3.

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