  Given a positive integer $u$ and a simple algebraic group $G$ defined over an
algebraically closed field $K$ of characteristic $p$, we derive properties
about the subvariety $G_{[u]}$ of $G$ consisting of elements of $G$ of order
dividing $u$. In particular, we determine the dimension of $G_{[u]}$,
completing results of Lawther [7] in the special case where $G$ is of adjoint
type. We also apply our results to the study of finite simple quotients of
triangle groups, giving further insight on a conjecture we proposed in [10] as
well as proving that some finite quasisimple groups are not quotients of
certain triangle groups.
