Permutations contained in transitive subgroups

TL;DR

This paper extends previous work to estimate the probability that a permutation has multiple fixed sets of specified sizes and uses this to determine the proportion of permutations in proper transitive subgroups of the symmetric group.

Contribution

It provides new estimates for the probability that permutations lie in transitive subgroups other than the full symmetric or alternating groups, including primitive and imprimitive cases.

Findings

Estimate for permutations in imprimitive transitive subgroups

Estimate for permutations in primitive transitive subgroups

Quantitative bounds on subgroup containment probabilities

Abstract

In the first paper in this series we estimated the probability that a random permutation $\pi\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\pi$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estimate for the proportion of permutations contained in a primitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$.