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arXiv · 2302.01770

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

Valentina Grazian·Carmine Monetta
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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.

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Valentina Grazian, Carmine Monetta. 2023-02-03. A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs. https://doi.org/10.1016/j.jalgebra.2023.07.002

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