Products of three conjugacy classes in the alternating group

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 Maróti [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 $c_1{c}_2{c}_3=g$.

43 pages