Let $a, b$ be elements of a NON-abelian group of orders $m, n$ respectively. What can you
say about the order of their product $ab$? In particular, I am interested in if we can assert that $mn$ divides the order of $ab$.

Question

Let $a, b$ be elements of a NON-abelian group of orders $m, n$ respectively. What can you 
say about the order of their product $ab$?  In particular, I am interested in if we can assert that $mn$ divides the order of $ab$.

anonymous · Answer

i think you might have it backwards, but i am not sure
for example you could have in $S_3$ an element of order 2 and an element of order 3, whose product has order 2

anonymous · Answer

one thing is for sure, both $m$ and $n$ divide the order of $G$ assuming of course that it is finite. and also $o(ab)$ divides the order of $G$