Let f : G H be a surjective homomorphism whose kernel is K. If S is a subgroup of H, then let S = { g G | f ( g ) S } .

(a) Prove that S is a subgroup of G that contains K .

(b) Let f denote the restriction of f to S . (That is, f : S H satisfies f ( g) = f ( g ) for all g S .) Prove that f is a homomorphism whose image is S and whose kernel is K .

(c) Hence, deduce that