TITLE: Can we find some constraint about order of $xy$ in a group $G$?
QUESTION [8 upvotes]: Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ?
I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case.
That is why I am looking for useful relation about $|xy|$.
I have found one which says that $|xy|=|yx|$ for any group.
Any useful relation about $|xy|$ would be appreciated, thanks.

REPLY [6 votes]: The only useful general statement I can think of, is that if $G$ is a finite solvable group, and $x,y \in G$ have coprime orders, then $o(xy),$ the order of $xy,$ is not coprime to $o(x)o(y).$ This is a statement equivalent to a Theorem of P. Hall (usually stated in the form that in a finite solvable group, the product of three elements of pairwise coprime order can't be the identity (except when they are all the identity)). This is relatively easy to prove: let $M$ be a minimal normal subgroup of $G$.  Since $M$ is an elementary Abelian $p$-group for some prime $p,$ the result is clear if $x,y \in M.$ Otherwise, by induction, the order of $xyM$ must have a common factor with $o(xM)o(yM).$ Since $o(xyM)$ divides $o(xy)$, we may conclude that $o(xy)$ certainly has a non-trivial common factor with $o(x)o(y).$
 If my memory is correct, the Theorem of P. Hall characterizes finite solvable groups, as a corollary of J.G. Thompson's $N$-group paper. That is to say, a finite group $G$ is solvable if  and only if it does not contain three distinct elements of pairwise coprime order whose product is the identity.