If G is an abelian group, a in G has order m, b in G has order n, and gcd(m,n)=1, show that ab has order mn. I am able to show that (ab)^(mn)=e actually occurs. I am having great difficulty however showing that mn is the smallest such integer. I tried to assume that there were a smaller integer, but I could not derive any contradiction. I tried to use the fact that gcd(m,n)=1 as best I could, but I can't make it work for me. Anyone have any ideas on where to go with this proof?