OD-characterization of the projective special linear groups \(L_2(q)\). (Q2909081)

For a finite group \(G\), let \(\pi_e(G)\) denote the set of element orders and \(\pi(G)\) denote the set of prime divisors of \(|G|\). The prime graph of \(G\) is a graph with vertex set \(\pi(G)\) and two distinct vertices \(p\) and \(q\) are joined by an edge, written \(p\thicksim q\), if and only if \(pq\in\pi_e(G)\).NEWLINENEWLINE If \(p\in\pi(G)\), then the degree of \(p\) is defined by \(\deg(p)=|\{q\in\pi(G)\mid p\thicksim q\}|\). If \(\pi(G)=\{p_1,p_2,\dots,p_k\}\), \(p_1<p_2<\cdots<p_k\), then we set \(D(G)=(\deg(p_1),\deg(p_2),\dots,\deg(p_k))\) and call it the degree pattern of \(G\). A group \(M\) is called \(k\)-fold OD-characterizable by the order and degree pattern if there exists exactly \(k\) non-isomorphic groups \(G\) such that \(|G|=|M|\) and \(D(G)=D(M)\). Moreover a \(1\)-fold OD-characterizable group is simply called an OD-characterizable group.NEWLINENEWLINE In Theorem 1.3(3)a of \textit{A. R. Moghaddamfar, A. R. Zokayi} and \textit{M. R. Darafsheh}, [Algebra Colloq. 12, No. 3, 431-442 (2005; Zbl 1072.20015)], the authors proved that the groups \(L_2(q)\) under certain conditions are OD-characterizable. In the paper under review the authors generalize this theorem to all the groups in the family \(L_2(q)\). This is the third family of groups proven to be OD-characterizable. Moreover, this result of the authors gives a positive answer to a conjecture of \textit{W.-J. Shi} for the groups \(L_2(q)\) [stated in Contemp. Math. 82, 171-180 (1989; Zbl 0668.20019)].