Products of three conjugacy classes in the alternating group

TL;DR

This paper proves that for large alternating groups, the product of three sufficiently large conjugacy classes covers the entire group, providing a constructive approach without character theory.

Contribution

It establishes that three large conjugacy classes in alternating groups multiply to the whole group, improving previous results and enabling a constructive algorithm.

Findings

Product of three large classes equals the group

Improves previous bounds with fewer classes

Constructive algorithm for element decomposition

Abstract

We prove that for $\delta$ small, $n$ large, and any three conjugacy classes $C_{1},C_{2},C_{3}$ of $G=\mathrm{Alt}(n)$ of size at least $|G|^{1-\delta}$ we have $C_{1}C_{2}C_{3}=G$. The result provides a positive answer to Problem 20.23 of the Kourovka Notebook [KM22], improves theorems of Garonzi and Mar\'oti [GM21] (using $4$ classes) and Rodgers [Rod02] (using larger classes), complements the known result for $G$ a simple group of Lie type [MP21] [LST24] [FM25], and is tight in several senses. Furthermore, since no character theory is involved, the proof can be used in principle to build a constructive algorithm that, given $g\in G$, outputs $c_{i}\in C_{i}$ such that $c_{1}c_{2}c_{3}=g$.