Cohen-Lenstra heuristics and random matrix theory over finite fields

Let g be a random element of a finite classical group G, and let \lambda_{z-1}(g) denote the partition corresponding to the polynomial z-1 in the rational canonical form of g. As the rank of G tends to infinity, \lambda_{z-1}(g) tends to a partition distributed according to a Cohen-Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition \lambda_{z-1}(g) and the Cohen-Lenstra type measure.