Let G be a finite group. (a) The centralizer of an element h ∈ G is the H = {g ∈ G | ghg−1 = h}. Show that H is a subgroup of G. (b) Let g, k ∈ G. Show that if ghg−1 = khk−1, then gH = kH, where H is as defined in (a). c) A conjugacy class is a set of the form Ch = {ghg−1| g ∈ G} for some h ∈ G. Use (b) to show that the number of elements in Ch divides the order of G. (d) If G has only two conjugacy classes. Show that |G| = 2.

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