Probability of generation by random permutations of given cycle type

TL;DR

This paper derives an approximate probability formula for the generation of the symmetric group by two random permutations with specified cycle structures, and applies it to show a 51% chance of generation from random conjugacy classes.

Contribution

The authors provide a new asymptotic probability formula for group generation by permutations with constrained cycle types, extending previous results.

Findings

Derived an explicit probability formula for transitivity and group generation.

Showed that two random elements from random conjugacy classes generate the group with about 51% probability.

Connected cycle structure constraints to group generation likelihoods.

Abstract

Suppose $\pi$ and $\pi'$ are two random elements of $S_n$ with constrained cycle types such that $\pi$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $\pi'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle \pi, \pi' \rangle$ is transitive and $G \geq A_n$ both have probability approximately \[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\] provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.