A characterization of the group \(\mathbb A_{p+3}\) by its non-commuting graph. (Q2920238)

Let \(G\) be a finite group. The non-commuting graph \(\nabla(G)\) of \(G\) is defined as follows: the set of vertices of \(\nabla(G)\) is \(G\setminus Z(G)\), where \(Z(G)\) is the center of \(G\), and two vertices are joined by an edge if and only if they do not commute.NEWLINENEWLINE It has been conjectured that if \(G\) and \(H\) are two non-Abelian finite groups such that \(\nabla(G)\cong\nabla(H)\), then \(|G|=|H|\) [see \textit{A. Abdollahi, S. Akbari} and \textit{H. R. Maimani}, J. Algebra 298, No. 2, 468-492 (2006; Zbl 1105.20016)].NEWLINENEWLINE In the paper under review, the authors show that if \(G\) is a finite group with \(\nabla(G)\cong\nabla(\mathbb A_{p+3})\) then \(G\cong\mathbb A_{p+3}\), where \(\mathbb A_{p+3}\) is the alternating group of degree \(p+3\), where \(p\) is a prime number.NEWLINENEWLINE In [\textit{M. R. Darafsheh}, Discrete Appl. Math. 157, No. 4, 833-837 (2009; Zbl 1184.20023)] it has been proved that the same result is valid for all finite simple groups with non connected prime graph. In this paper the main theorem does not depend on the connectedness of the prime graph of \(\mathbb A_{p+3}\), because \(\mathbb A_{p+3}\) has disconnected or connected prime graph depending on \(p\).NEWLINENEWLINE In order to prove the main result the authors use also some results of E. Artin (1955) and the classification of finite simple groups.