A generalization of the probability that the commutator of two group elements is equal to a given element

TL;DR

This paper extends the concept of the probability that the commutator of two group elements equals a specific element, exploring broader contexts and structural restrictions of the group.

Contribution

It generalizes previous notions by considering wider contexts and provides structural restrictions on groups where this probability is studied.

Findings

Identifies structural restrictions on groups based on commutator probabilities

Extends the concept to broader algebraic contexts

Provides new insights into the relationship between group structure and commutator probabilities

Abstract

The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with combinatorial techniques. Here we illustrate that a wider context may be considered and show some structural restrictions on the group.