You're given $k$ permutations $a_1,\dots,a_k$. Consider closure of this set under the composition operation. What are most efficient and simple algorithms to calculate the size of this closure?
The most efficient (and also simplest) algorithms for this are based on the notion of a strong generating set, introduced by Sims. Strong generating sets can be computed efficiently using the Schreier-Sims algorithm. Essentially any introductory book that talks about computational group theory should have a decent introduction to this (and if it contains an introduction to this that isn't decent, I'd think it's not a very good book). While the standard deterministic algorithm for this takes quadratic time, there is a randomized nearly-linear time algorithm. See Seress's book "Permutation Group Algorithms," Chapter 4 (the nearly-linear time Monte Carlo algorithm is Theorem 4.5.5).