JOURNAL OF
ALGEBRAIC
COMBINATORICS

Let G be an adjoint simple algebraic group over an algebraically closed field of characteristic p; let Φ  be the root system of G, and take t\in \Bbb N. Lawther has proven that the dimension of the set G _{[ t]}={ g\in  G: g ^t =1} depends only on Φ  and  t. In particular the value is independent of the characteristic p; this was observed for t small and prime by Liebeck. Since G _{[ t]} is clearly a disjoint union of conjugacy classes the question arises as to whether a similar result holds if we replace G _{[ t]} by one of those classes. This paper provides a partial answer to that question. A special case of what we have proven is the following. Take p, q to be distinct primes and G _p and G _q to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic p and q respectively. If s\in  G _p has order q then there exists an element u\in  G _q such that o( u)= o( s) and = s^G_p \dim u^{G_{q}}=\dim s^{G_{p}} .