Let G be a group and let a e G. The set CG(a) = {x € G | xa = that commute with a is called the Centralizer of a in G. The set Z(G) = {x € G | xa = ax for all a e G} of all elements of G that commute with every element of G is called the Center of G. a.x} of all elements (a) Prove that Z(G) is a subgroup of G. (b) Prove that Cg(a) is a subgroup of G. (c) Compute Cg(a) when G = S3 and a = (1, 2).

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