46 Chapter 1. Groups I ∗ 1.59. (i) Prove that the intersection of any family of normal subgroups of a group G is itself a normal subgroup of G. (ii) If X is a subset of a group G, let N be the intersection of all the normal subgroups of G containing X. Prove that X ⊆ N G, and that if S is any normal subgroup of G containing X, then N ⊆ S. We call N the normal subgroup of G generated by X. (iii) If X is a subset of a group G and N is the normal subgroup generated by X, prove that N is the subgroup generated by all the conjugates of elements in X. ∗ 1.60. If K G and K ⊆ H ⊆ G, prove that K H. ∗ 1.61. Define W = (1 2)(3 4) , the cyclic subgroup of S4 generated by (1 2)(3 4). Show that W is a normal subgroup of V, but that W is not a normal subgroup of S4. Conclude that normality is not transitive: W V and V G do not imply W G. ∗ 1.62. Let G be a finite abelian group written multiplicatively. Prove that if |G| is odd, then every x ∈ G has a unique square root that is, there exists exactly one g ∈ G with g2 = x. Hint. Show that squaring is an injective function G → G. 1.63. Give an example of a group G, a subgroup H ⊆ G, and an element g ∈ G with [G : H] = 3 and g3 / ∈ H. Compare with Proposition 1.68(i). Hint. Take G = S3, H = (1 2) , and g = (2 3). ∗ 1.64. Show that the center of GL(2, R) is the set of all scalar matrices aI with a = 0. Hint. Show that if A is a matrix that is not a scalar matrix, then there is some nonsingular matrix that does not commute with A. [The generalization of this to n × n matrices is true see Corollary 2.133(ii)]. ∗ 1.65. Prove that every isometry in the symmetry group Σ(πn) permutes the vertices {v1,. . . , vn} of πn. (See FCAA, Theorem 2.65.) ∗ 1.66. Define A = ζ 0 0 ζ−1 and B = [ 0 1 i 0 ], where ζ = e2πi/n is a primitive nth root of unity. (i) Prove that A has order n and B has order 2. (ii) Prove that BAB = A−1. (iii) Prove that the matrices of the form Ai and BAi, for 0 ≤ i n, form a multiplicative subgroup G ⊆ GL(2, C). Hint. Consider cases AiAj , AiBAj , BAiAj , and (BAi)(BAj ). (iv) Prove that each matrix in G has a unique expression of the form BiAj , where i = 0, 1 and 0 ≤ j n. Conclude that |G| = 2n. (v) Prove that G ∼ = D2n. Hint. Define a function G → D2n using the unique expression of elements in G in the form BiAj . ∗ 1.67. Let Q = { I, A, A2,A3,B, BA, BA2,BA3 }, where A = 0 1 −1 0 and B = [ 0 i i 0 ]. (i) Prove that Q is a nonabelian group with binary operation matrix multiplication. (ii) Prove that A4 = I, B2 = A2, and BAB−1 = A−1.

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