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Suggested Exercises II Symmetry & Groups 1. Let G be a group. The center of G, denoted Z(G), is the set of all elements g ∈ G such that for every h ∈ G, hg = gh. Prove Z(G) is a normal subgroup of G. 2. Let G = S3 , the symmetric group of degree 3, and H = {e, (1 2)}. (a) Write down all the left cosets of H in G. (b) Write down all the right cosets of H in G. (c) Is every left coset of H a right coset of H? (d) Repeat the problem with H 0 = {e, (1 2 3), (1 3 2)}. 3. If G is a group and N C G, prove G/N is abelian if and only if aba−1 b−1 ∈ N for all a, b ∈ G. 4. Let G be the group of all functions f : [0, 1] → R, where we define addition for f, g ∈ G by (f + g)(x) = f (x) + g(x) for every x ∈ [0, 1]. If N = {f ∈ G| f ( 41 ) = 0}, prove G/N ' R under +. Hint: Use the first homomorphism theorem. 5. Let Sn be the symmetric group on n letters. A cycle in Sn is even if it can be written as a product of an even number of transpositions, and is odd otherwise. For example, if n = 3, (1 2 3) = (1 2)(2 3) is even, and (1 3) is odd. Let An be the subset of all even cycles in Sn . (a) Prove An is a subgroup of Sn . (b) Prove An is normal in Sn . Hint: Try to find a homomorphism φ : Sn → {−1, 1}, under multiplication, which has An as the kernel. 1