Prove that if N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal subgroup of H.

Prove that if H is a normal subgroup of G of prime index p then for all K < G either (1) K < H or (ii) G = HK and |K : K n H|= p.

Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.

Let H be a subgroup of Sn. (a) Show that either all the permutations in H are even, or else half the permutations in H are even and half are odd. (b) Show that the set of even permutations in H form a subgroup of H.