conjugationGroup actions provide a useful language to define several important but somewhat technical notions in group theory with minimal fuss, as in the next two examples. A common hint for both parts is to consider Gx and GX. (a) Let X = G (considered as a set) and consider the action g(x) = gxg1 for g G and x X as in Problem 6(a). Define: the centralizer CG(x) of x in G by CG(x) := {g G | gx = xg}; and the center Z(G) of G by Z(G) := {g G | gx = xg for every x X}. Show that Z(G) G, and that, for every x X, Z(G) CG(x) and CG(x) G. (b) Let X = {H G}, the set of subgroups of G, and consider the conjugation action g(H) = gHg1 of G on X. Define: the normalizer NG(H) of H in G by NG(H) := {g G | gHg1 = H}. Show that NG(H) G and that H NG(H)