A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

TL;DR

The paper investigates whether the non-commuting graph $\Gamma_G$ of a finite group determines nilpotency, focusing on the class of AC-groups. It introduces Conjecture 3, showing that together with $|Z(G)| \ge |Z(H)|$ this implies Conjecture 2, and proves this for finite AC-groups by analyzing centralizers via graph-theoretic constraints. The main result proves that if $G$ is a finite non-abelian nilpotent AC-group and $\Gamma_G \cong \Gamma_H$ with $|Z(G)| \ge |Z(H)|$, then $H$ is nilpotent, representing a substantial advance toward understanding how much information about group structure is captured by non-commuting graphs. This work bridges graph-theoretic invariants and group-theoretic structure, informing broader questions about the detectability of nilpotency from non-commuting graphs.

Abstract

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3) that, together with the assumption $|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.

Theorems & Definitions (31)

• Conjecture 1
• Conjecture 2
• Theorem 1.1
• Theorem 1.2
• Conjecture 3
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Lemma 2.2
• Corollary 2.3