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

Wei Zhou

TL;DR

The paper investigates whether a finite group $G$ that shares the same set of conjugacy-class sizes ${\mathrm{N}}(G)$ with a nilpotent group must be nilpotent. It presents an explicit construction via a semidirect product $G=H\rtimes (A\rtimes B)$ with $H$ abelian and carefully chosen $A,B$, and computes conjugacy-class-size indices to show ${\mathrm{N}}(G)={1,p,p^{p-2}}\times{1,q}$ while the center is trivial, ${\mathrm{Z}}(G)=1$. The resulting centerless group has the same conjugacy-class-size spectrum as a nilpotent group, demonstrating that sharing ${\mathrm{N}}(G)$ does not force nilpotency and answering Camina's 2006 question in the negative. This highlights limitations of conjugacy-class-size data for deducing nilpotent structure in finite groups.

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.

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

TL;DR

The paper investigates whether a finite group that shares the same set of conjugacy-class sizes with a nilpotent group must be nilpotent. It presents an explicit construction via a semidirect product with abelian and carefully chosen , and computes conjugacy-class-size indices to show while the center is trivial, . The resulting centerless group has the same conjugacy-class-size spectrum as a nilpotent group, demonstrating that sharing does not force nilpotency and answering Camina's 2006 question in the negative. This highlights limitations of conjugacy-class-size data for deducing nilpotent structure in finite groups.

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.
Paper Structure (3 sections, 3 theorems)

This paper contains 3 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof the main theorem

Key Result

Corollary 1

Let $p$ and $q$ be primes such that $p = 2q + 1$. Let $G$, $H$ and $A$ be as defined above. Let $L = P \times Q$, where $P = H \rtimes A$ and $Q = C_{q^2}\rtimes C_q$. Then, we have ${\mathrm{N}}(G) = {\mathrm{N}}(L)$.

Theorems & Definitions (5)

  • Corollary
  • Lemma 1
  • proof
  • Lemma 2
  • proof