Aspects of the conjugacy class structure of simple algebraic groups. (Q968249)

Let \(G\) be an adjoint simple algebraic group over an algebraically closed field of characteristic \(p\), \(\Phi\) the root system of \(G\), and \(t\in\mathbb{N}\). Lawther has proven that the dimension of the set \(G_{[t]}=\{g\in G:g^t=1\}\) depends only on \(\Phi\) and \(t\). In particular the dimension is independent of \(p\); which was observed for small \(t\) by Liebeck. Since \(G_{[t]}\) is clearly a disjoint union of conjugacy classes, the question arises whether a similar result holds if we replace \(G_{[t]}\) by one of those classes. This paper provides a partial answer to that question. E.g., 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 \(u\in G_q\) of the same order such that \(\dim u^{G_q}=\dim s^{G_p}\).