Product decompositions of quasirandom groups and a Jordan type theorem

2007/03/12 by Nikolay Nikolov, Nikolov, Nikolay, László Pyber +1 · 1 citation
Engineering · Mathematics · #20D06 #20F69 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.math/0703343

openalex publication_date 2007/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with |B| > |G| / k^1/3 we have B³ = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck² for some absolute constant c, hence a product-free subset of size at least |G| / c'k. This answers a question of Gowers.

Citations

• Sum-free sets in abelian groups

Cited by

• Products of three conjugacy classes in the alternating group

Related

• Quasirandom groups
• Product-free subsets of groups, then and now
• Generalizations of product-free subsets
• On the proportion of p-elements in a finite group, and a modular Jordan type theorem
• On the product decomposition conjecture for finite simple groups