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I recently came across the claim that if is a cyclic group and are subgroups with orders respectively, then the order of is .
I first tried to prove that was cyclic in the hope that it would make things easier. This wasn't too difficult. I first proved that any subgroup of a cyclic group is cyclic, then the fact followed since , being the intersection of two subgroups is itself a subgroup.
I then showed that if is generated by , and n,m are the smallest positive integers such that generate and respectively then the smallest power of that generates was equal to the lowest common multiple of and , but this approach has got me nowhere.