Gaps in probabilities of satisfying some commutator-like identities

TL;DR

This paper establishes a probabilistic gap for finite groups satisfying certain commutator-like identities, showing the probability is either 1 or bounded away from 1 by a positive constant.

Contribution

It proves a new probabilistic gap result for the satisfaction of specific commutator identities in finite groups.

Findings

Probability of satisfying the identities is either 1 or at most a constant less than 1.

Identifies a positive constant δ < 1 as a threshold.

Provides bounds on the likelihood of certain algebraic identities in finite groups.

Abstract

We show that there is a positive constant $\delta < 1$ such that the probability of satisfying either the $2$-Engel identity $[X_1, X_2, X_2] = 1$ or the metabelian identity $[[X_1, X_2], [X_3, X_4]] = 1$ in a finite group is either $1$ or at most $\delta$.