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

Source: [Probability that a given element of a group is a commutator of any two randomly chosen group elements](<https://www.sota2.com/research/paper/68fa54fb1e209131ce99007d>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T13:18:36.524Z

Authors: Rajat K. Nath, Manoj K. Yadav

Published: 2012-12-18T11:08:10.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1212.4306>)

[PDF](<https://arxiv.org/pdf/1212.4306.pdf>)