Suppose that in a group G we have o(g) = m and o(h) = n with gcd (m, n) = 1.

(1) If gh = hg, show that o(gh) = mn.

(2) If a G with o(a) = mn, show that there exists g and h in G such that a = gh = hg et o(g) = m et o(h) = n.

Note : o(g) = order of g