# Random generation with cycle type restrictions

## TL;DR

This paper investigates the probability that two permutations with certain cycle restrictions generate the alternating group, providing new insights into random generation in symmetric groups under cycle type constraints.

## Contribution

It establishes conditions under which permutations with restricted cycle types are likely to generate the alternating group, extending understanding of random generation with cycle restrictions.

## Key findings

- Permutations with limited fixed points and 2-cycles tend to generate at least A_n.
- For integers m with divisors d in 3 to o(n^{1/2}), two random elements of order m likely generate A_n.
- Provides conditions for when two elements of a given order generate the alternating group.

## Abstract

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi, \pi' \in S_n$, we prove that $\pi$ and a random conjugate of $\pi'$ are likely to generate at least $A_n$ provided only that $\pi$ and $\pi'$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the following question: For which positive integers $m$ should we expect two random elements of order $m$ to generate $A_n$? Among other things, we give a positive answer for any $m$ having any divisor $d$ in the range $3 \leq d \leq o(n^{1/2})$.