On the proportion of permutations of order a multiple of the degree

We study permutations of a set of size n for which the order is a multiple of n. We prove that, for large n, most such elements lie in one of two families. The first family consists of those permutations with a single very large cycle of order dividing n and includes the n-cycles, and the second consists of permutations for which the cycles of length dividing n have total length significantly less than n. This work was inspired by the algorithmic problem of fast recognition of large symmetric groups acting primitively on subsets.