Asypmtotic Behaviour of the Conjugacy Probability of the Alternating Group

TL;DR

This paper investigates the asymptotic behavior of the conjugacy probability in the alternating group, showing that it scales with 1/n^2 and converges to a specific constant as n grows large.

Contribution

It extends previous methods to explicitly determine the limit of n^2 times the conjugacy probability in A_n as n approaches infinity.

Findings

n^2 * κ(A_n) converges to a constant A as n→∞

Provides an explicit value for the constant A

Extends existing methods to analyze asymptotic conjugacy probabilities

Abstract

For $G$ a finite group, let $\kappa(G)$ be the probability that $\sigma,\tau\in G$ are conjugate, when $\sigma$ and $\tau$ are chosen independently and uniformly at random. In this paper, we extend the methods of Blackburn et al [2012] to show that $n^2\kappa(A_n) \to A$ as $n\to\infty$ for a constant $A$, which we will give explicitly.