1. Let G be a group, with subgroup H G. Let G Sym(G/H) be the left-coset action, defined by (g)(aH) = gaH. Show that ker() = gG gHg1. 2. Let G be group with subgroup H G, and let K = ker() as in the previous problem. Suppose H has finite index in G, and write m = [G H]. Show that G/K is isomorphic to a subgroup of Sym(G/H). Use this to show that [G K] divides m !, and then use this to show that [H K] divides (m 1)