Products of three conjugacy classes in the alternating group
Daniele Dona
TL;DR
The paper resolves Kourovka Notebook Problem 20.23 for the alternating group by proving that, for small δ and large n, any three conjugacy classes with size at least |G|^{1−δ} multiply to the whole group, i.e., C1C2C3=G, with an intermediate containment C1C2⊇C3 guiding the proof. The approach is elementary and constructive, encoding conjugacy classes via class strings that capture cycle structure and performing a sequence of seven reductions to reduce to a Sym(n) framework while preserving parity and alignment; witnesses for c1c2=c3 are then lifted back to the original Alt(n). The method yields a constructive algorithm, should one wish to output c_i in C_i with c1c2c3=g for a given target g, and it complements analogous results for Lie-type simple groups by providing a parallel elementary route in the alternating family. In addition to solving the problem in Alt(n), the work clarifies tightness and establishes a robust framework for analyzing products of large conjugacy classes in finite simple groups, with potential extensions to normal subsets and related settings.
Abstract
We prove that for $δ$ small, $n$ large, and any three conjugacy classes $C_{1},C_{2},C_{3}$ of $G=\mathrm{Alt}(n)$ of size at least $|G|^{1-δ}$ 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 $c_{1}c_{2}c_{3}=g$.