1. (2 points) Suppose G is a group, and H is a subgroup of index 2. Prove that H is a normal subgroup of G. (Hint: start by proving aH = Ha for any a E G.) 2. (3 points) Let G = Sm, the symmetric group on n letters, for some positive integer n. Suppose (az az • Am) is a cycle in Sn (so the ai are necessarily distinct). Prove that for any o ESR o(aja2 am)o-1 = (o(au) o(az) olam))