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Let G be a group and let a ∈ G. The set CG(a) = {x ∈...

Let G be a group and let a ∈ G. The set CG(a) = {x ∈ G | xa = ax} of all elements that commute with a is called the Centralizer of a in G.

(b) Compute CG(a) when G = S3and a = (1, 2).

(c) Compute CG(a) when G = S4 and a = (1, 2).

(d) Prove that Z(G) = ∩a∈GCG(a).

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