Products of three conjugacy classes in the alternating group

2025/05/09 by Daniele Dona, Dona, Daniele
Mathematics · #20B30 #20E45 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2505.06012

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for δ small, n large, and any three conjugacy classes C₁,C₂,C₃ of G=Alt(n) of size at least |G|^1-δ we have C₁C₂C₃=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∈ G, outputs c_i∈ C_i such that c₁c₂c₃=g.

Citations

• On the Gowers trick for classical simple groups
• Bounds for Characters of the Symmetric Group: A Hypercontractive Approach
• Squares of Conjugacy Classes in Alternating Groups
• Alternating groups as products of cycle classes - II
• Alternating groups as products of four conjugacy classes
• A generalization of the diameter bound of Liebeck and Shalev for finite\n simple groups
• Product-free subsets of groups, then and now
• Product decompositions of quasirandom groups and a Jordan type theorem
• Character Theory of Symmetric Groups, Subgroup Growth of Fuchsian Groups, and Random Walks

Related

• A lower-bound for the number of conjugacy classes of A_n
• Alternating groups as products of four conjugacy classes
• Products of conjugacy classes of the alternating group
• On Thompson's conjecture for alternating group of large degree
• Squares of Conjugacy Classes in Alternating Groups