Probability that a given element of a group is a commutator of any two randomly chosen group elements

TL;DR

This paper investigates the probability that a specific element in a group's commutator subgroup is a commutator of two randomly selected elements, providing explicit formulas for certain groups with limited conjugacy class sizes.

Contribution

It offers explicit probability formulas for elements being commutators in groups with only two conjugacy class sizes, and re-proves a classification result for groups with conjugacy class sizes {1, p}.

Findings

Derived explicit formulas for probabilities in specific groups.

Re-proved that groups with conjugacy class sizes {1, p} are isoclinic to extraspecial p-groups.

Analyzed groups with exactly two conjugacy class sizes.

Abstract

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes. We re-prove the fact that if $G$ is a finite group such that the set of its conjugacy class sizes is $\{1, p\}$, where $p$ is a prime integer, then $G$ is isoclinic (in the sense of P. Hall) to an extraspecial $p$-group.