Some asymptotic results on finite vector spaces

AbstractIn this paper, we apply the vector space cycle index derived by Kung (Linear Algebra Appl. 36 (1981), 141–155) to derive asymptotic expansions of a number of quantities for GL(d, q) and Mat(d, q) as d → ∞, extending to these groups the statistical group theory of Erdös and Turan (Acta Math. Acad. Sci. Hungar.19 (1968), 413–435). Among the quantities we will consider are the number of orbits under the action of GL(d, q) by conjugacy, the fraction of matrices that are diagonalizable, the expected number of distinct polynomials occurring in the rational canonical form (a generalization of the expected number of cycles for the permutation group) and the expected degree of the rth highest degree polynomial