  Probabilistic algorithms are applied to prove theorems about the finite
general linear and unitary groups which are typically proved by techniques such
as character theory and Moebius inversion. Among the theorems studied are
Steinberg's count of unipotent elements, Rudvalis and Shindoda's work on the
fixed space of a random matrix, and Lusztig's work on counting nilpotent
matrices of a given rank.
