On the probability of satisfying a word in nilpotent groups of class 2

TL;DR

This paper proves that in finite nilpotent groups of class 2, the probability that a random n-tuple satisfies a given word is at least 1 divided by the group's order, confirming a special case of a conjecture.

Contribution

It establishes a lower bound on the probability of satisfying a word in finite nilpotent groups of class 2, addressing a specific conjecture in group theory.

Findings

Probability is at least 1/|G| for such groups.

Confirms a special case of Alon Amit's conjecture.

Provides insight into word satisfaction in nilpotent groups.

Abstract

Let G be a finite group of nilpotency class 2 and w a group word. In this short paper we show that the probability that a random n-tuple of elements from G satisfies w is at least one over the order of G. This answers a special case of a conjecture of Alon Amit.