On commuting probabilities in finite groups and rings

TL;DR

This paper explores the relationship between commuting probabilities in finite rings and finite nilpotent groups, establishing their equality under certain conditions and conjecturing their overall equivalence.

Contribution

The paper proves that the set of commuting probabilities in finite rings is contained within that of finite nilpotent groups of class ≤2, and shows they are equal for groups and rings with odd order.

Findings

Commuting probabilities in finite rings are a subset of those in finite nilpotent groups of class ≤2.

Equality of these sets is proven for groups and rings with odd number of elements.

The authors conjecture the sets are equal in general.

Abstract

We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class $\le2$. We believe that these two sets are equal; we prove they are equal, when restricted to groups and rings with odd number of elements.