Consider the number of fixed points in a permutation chosen uniformly at random from the symmetric group on $n$ elements - this gives a probability distribution. For $k < n$, the $k$-th moments of this distribution are the same as the Poisson distribution with $\lambda = 1$.

Since you can't pick uniformly randomly from an infinite set, it doesn't make sense to ask if you get exactly the Poisson distribution in the limit of the symmetric group on a countably infinite set. But can you get there in another way? Is there a known way to take limits in some category of distributions and groups that leads to the Poisson distribution for the fixed points of some countably infinite group of permutations? Alternately, is there a known structure that can be put on $|S_\omega|$ that gives rise to a Poisson process that is analogous to the finite case?

Also, are there other infinite families of finite permutation groups that give rise to a similarly elegant characterization of the distribution of fixed points? Say by taking one of the families of finite simple groups and applying the same finite extension to each of them?

(Please excuse me if I accidentally posted this twice - I didn't see the first attempt go through.)