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

Let G_F 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 U_F-conjugacy class, U_F a p-Sylow subgroup of G_F, is a q-power, where q is the level of F . We also determine the existence of power series inducing isomorphisms between U_F and its Lie algebra.