(1) Prove o(h(a)) < ∞.
(2) Prove o(h(a)) | o(a).
Note: o(a) denotes the order of a, and o(h(a)), denotes the order of h(a). Also, in (2), what o(h(a)) | o(a) means, is that o(h(a) divides o(a).
(The more helpful, the better the appreciation.)
Update:Oh excuse me, it is supposed to say, "in which G and H are groups".
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Answers & Comments
(1) is easy, once you have that if e is the identity in G, and 1 is the identity in H, that h(e) = 1. Just look at h(a^n).
(2) Let n = o(h(a)) and m = o(a). Compare a^n and a^m. If n doesn't divide m, what does that say about h(a^n)?