Probabilistically nilpotent groups of class two

TL;DR

This paper characterizes finite groups with a positive lower bound on the probability that three randomly chosen elements satisfy a specific commutator relation, revealing their structure involves bounded class-4 nilpotent subgroups.

Contribution

It establishes structural results for finite groups with a positive probability of certain commutator identities, linking probabilistic properties to algebraic hierarchy.

Findings

Finite groups with $d_2(G) \\geq \\epsilon$ have a class-4 nilpotent subgroup with bounded index and derived subgroup size.

Infinite groups with bounded conjugacy of commutators are finite-by-class-3-nilpotent-by-finite.

Provides a structural classification based on probabilistic commutator conditions.

Abstract

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon > 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G : H] $ and $|\gamma_4(H)|$ are both bounded in terms of $\epsilon$. We also show that if $G$ is an infinite group whose commutators have boundedly many conjugates, or indeed if $G$ satisfies a certain more general commutator covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.