  Picking permutations at random, the expected number of k-cycles is known to
be 1/k and is, in particular, independent of the size of the permuted set. This
short note gives similar size-independent statistics of finite general linear
groups: ones that depend only on small minors. The proof technique uses
combinatorics of categories, motivated by representation stability, and applies
simultaneously to symmetric groups, finite linear groups and many other
settings.
