On Groups with the Same Set of Conjugacy Class Sizes as Nilpotent Groups

TL;DR

This paper constructs groups with identical conjugacy class size sets as nilpotent groups but with trivial centers, answering a question from 2006 about the relationship between conjugacy class sizes and group structure.

Contribution

It provides explicit examples of non-nilpotent groups sharing conjugacy class size sets with nilpotent groups, addressing a longstanding open question.

Findings

Existence of non-nilpotent groups with conjugacy class sizes matching nilpotent groups

Construction of groups with trivial centers and identical conjugacy class sets

Resolution of a question posed by A. R. Camina in 2006

Abstract

We construct examples of groups which have the same set of conjugacy class sizes as nilpotent groups, while their center is trivial. This answers a question posed by A. R. Camina in 2006.