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Homework 9 Due Wednesday, April 7 1. Let G = Z18 and H = h [6] i. What is the order of 5 + H in G/H? 2. (a) Let G be a group that is not abelian. Prove that G/Z(G) is not cyclic. (b) Prove that if G is a group with |G| = pq where p and q are prime numbers, then either G is commutative or Z(G) = {1}. 3. Let C× be the group of nonzero complex numbers under multiplication. Let S 1 = {z ∈ C× : |z| = 1} be the circle group with the operation of multiplication. Prove that S 1 / C× and C× /S 1 ∼ = R+ , where R+ denotes the group of positive real numbers under multiplication. 4. Let G = R× be the group of nonzero real numbers under multiplication. Let N = {−1, 1}. Prove that N is a normal subgroup of G and that G/N is isomorphic to R+ , the group of positive real numbers under multiplication. 5. Let G be a group. For each g ∈ G let γg : G → G by γg (x) = gxg −1 . Let Inn(G) = {γg : g ∈ G}. (a) Prove that Inn(G) is a group under composition. (We call Inn(G) the group of inner automorphisms of G.) (b) Prove that Inn(G) ∼ = G/Z(G). (c) If D2n is the group of symmetries on the regular n-gon, then how many elements are in Inn(D2n )? 6. An element a of a group G is said to be a square if a = b2 for some b ∈ G. Let G be an abelian group and let H be a subgroup of G. If every element of H is a square, and every element of G/H is a square, then prove that every element of G is a square. 7. Let G be a finite group and let H / G. (a) Prove that the order of the element gH in G/H must divide the order of the element g in G. (b) Prove that if x ∈ G and gcd(|x|, |G/H|) = 1, then x ∈ H. 8. Prove that (a) U (Z9 ) ∼ = Z6 (b) U (Z15 ) ∼ = Z4 × Z2