Finite $p$-Groups of Nilpotency Class $3$ with Two Conjugacy Class Sizes

TL;DR

This paper characterizes finite p-groups of nilpotency class 3 with exactly two conjugacy class sizes, proving existence conditions and uniqueness up to isoclinism for certain parameters.

Contribution

It establishes the existence criterion and uniqueness (up to isoclinism) for finite p-groups of class 3 with two conjugacy class sizes, specifically when the size is 1 and p^n with n even.

Findings

Such groups exist if and only if n is even

For each even n, the group is unique up to isoclinism

Provides a classification of these p-groups based on conjugacy class sizes

Abstract

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).