A subgroup HGis normal if gH= Hg for all gG. (a) Prove that H is normal if and only if g1Hg= H for all gG, where g1Hg= [g1hg: hH]. (b) Prove that if G is abelian, then every subgroup of G is normal. (c) Prove that (1 2)is not a normal subgroup of S3. (d) Prove that if f: GH is a group homomorphism, then ker f is a normal subgroup of G.