Let o be a homomorphism from a group G to a group H and let U ≤G be a subgroup of G. Show that
(a) o(U) = {o(u) | u EU} is a subgroup of H.
(b) If U is cyclic, then (U) is cyclic.
(c) If U is abelian, then o(U) is abelian. The converse is not true.
(d) If U is normal in G, then o(U) is normal in (G).
(e) If |Keron, then o is an n-to-one mapping from G to o(G).
(f) If V is a subgroup of H, then o¹(V) = {g €G | o(g) € V} is a subgroup of G.
(g) If N is a normal subgroup of H, then o-¹(N) = {g Go(g) E N} is a normal subgroup of G.
(h) Ifo is onto and Kero = {e}, then o is an isomorphism from G to H.
(i) Ifo is onto then the mapping from G/Kero to H, given by gKeroo(g), is an isomor- phism from G/Kero to H.