Centralizers and conjugacy. Let G be a group. For any element a EG, the centralizer Ca of a is the set of all elements which commute with a: Ca = {x € G: xa = ax} = {x € G:a= xax1}. (1) Show that Ca is a subgroup of G. (2) Show that xax-1 = yay-1 if and only if xCa =yCa. (3) Show that the conjugates of a are in bijective correspondence with the left cosets of Ca. Thus if G is finite, the conjugacy class [a] has [G: Ca] elements, and the size of the conjugacy class divides the order of the group. (4) Let p be prime and let G be a group of order pa for some a > 1. Recall that the center of G is C(G) = {a EG: ax = xa for all x EG}. Show that C(G) + {e}. (Hint: Use problem