Permutations contained in transitive subgroups

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 ${\mathcal{A}}_n$.

36 pages, 1 figure. Reformatted for Discrete Analysis but otherwise identical to the previous version

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