Let o be a homomorphism from a group G to a group H and let g € G be an element of G. Let [g] denote the order of g. Show that
(a) o takes the identity of G to the identity of H.
(b) o(g") = o(g)" for all n € Z.
(c) If g is finite, then lo(g)] divides g.
(d) Kero = {g Go(g) = e) is a subgroup of G (here, e is the identity element in H).
(e) o(a)= o(b) if and only if aKero=bKero.
(f) If o(g) = h, then o-¹(h) = {re Go(x)=h} = gKero.