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

TL;DR

This paper investigates whether the structure of non-commuting graphs of finite groups can determine their nilpotency, proposing a new conjecture and proving it for certain classes of groups.

Contribution

It introduces a new conjecture linking non-commuting graphs and nilpotency, and proves it for groups with abelian centralizers of non-central elements.

Findings

Proposed a new conjecture relating non-commuting graphs to nilpotency.

Proved the conjecture for groups with abelian centralizers of non-central elements.

Established conditions under which the non-commuting graph determines nilpotency.

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.