  The asymptotic study of the conjugacy classes of a random element of the
finite affine group leads one to define a probability measure on the set of all
partitions of all positive integers. Four different probabilistic
understandings of this measure are given--three using symmetric function theory
and one using Markov chains. This leads to non-trivial enumerative results.
Cycle index generating functions are derived and are used to compute the large
dimension limiting probabilities that an element of the affine group is
separable, cyclic, or semisimple and to study the convergence to these limits.
This yields the first examples of such computations for a maximal parabolic
subgroup of a finite classical group.
