when do two elements generate a p-group?

Question

Consider the symmetric group $S_n$, and let $x,y \in S_n$ such that $x$ has order $p^k$, and y has order $p^l$ for some prime $p$, and positive integers $k,l$. Can we give any kind of precise conditions on $x,y$ to ensure that the group $<x,y>$ is a $p$-group? Of course, if you fix a 2-generated $p$-group, and impose the relations in its presentation as conditions on $x,y$, then you definitely get a p-group. Or, same when $x$ and $y$ commute. But, my question is not quite that. I was wondering if there are more general conditions (without fixing any $p$-group in mind) that make $<x,y>$ a $p$-group for sure.

Answer 1

If $xy=yx$ then it will be a $p$-group.