A characterization of the finite simple group \(L_{16}(2)\) by its prime graph.
Let \(G\) be a finite group. The prime graph of \(G\), denoted by \(\Gamma(G)\), is a simple graph whose vertex set is the set of all prime divisors \(\Pi(G)\) of \(|G|\), and two distinct vertices \(p,q\) belonging to \(\Pi(G)\) are joined by an edge if \(G\) has an element of order \(pq\). The group \(G\) is said to be recognizable by its prime graph if for every group \(H\) with \(\Gamma(G)=\Gamma(H)\) we have \(G\cong H\). In the paper under review the authors prove that the projective special linear group \(\text{PSL}_{16}(2)\) is recognizable by its prime graph. As a consequence of the main result of this paper the authors deduce that the group \(L_{16}(2)\) is recognizable by the set of its element orders. But it is noteworth to mention that \textit{M. A. Grechkoseeva, W. Shi} and \textit{A. V. Vasilev} [Algebra Colloq. 14, No. 4, 585-591 (2007; Zbl 1139.20012)] proved that the groups \(\text{PSL}_{16}(2^m)\) are recognizable by their sets of element orders. Also \textit{A. V. Zavarnitsine} and \textit{V. D. Mazurov} [Dokl. Math. 74, No. 1, 569-572 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 409, No. 6, 736-739 (2006; see the following review Zbl 1143.20010)] proved that the groups \(\text{PSL}_n(2)\), \(n\geq 3\), are recognizable by their sets of element orders. Therefore the conjecture mentioned in the paper is answered affirmatively. I would like to mention that this conjecture first was put forward by \textit{M. R. Darafsheh} and \textit{A. R. Moghaddamfar} [Commun. Algebra 29, No. 1, 465-475 (2001; Zbl 0989.20018)].
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