Global Cayley maps and conjugacy class sizes of maximal unipotent subgroups of finite simple groups of Lie type

Let GF be the group of points fixed by the Frobenius endomorphism F of a simple algebraic group G defined in characteristic p. In this paper, we show that if p is a good prime then the size of every UF-conjugacy class, UF a p-Sylow subgroup of GF, is a q-power, where q is the level of F. We also determine the existence of power series inducing isomorphisms between UF and its Lie algebra.