Simple groups which are 2-fold OD-characterizable.
The well-known notion of the prime graph \(\Gamma(G)\) of a finite group \(G\) has many important applications to the study of arithmetical properties of finite groups. In particular, results on the prime graph of finite simple groups are used in the investigations of the problem of the recognition of finite almost simple groups by their orders and some properties of their prime graphs. Let \(G\) be a finite group and \(\pi(|G|)=\{p_1,\dots,p_k\}\) with \(p_1<p_2<\cdots<p_k\). For \(p\in\pi(|G|)\), let \(\deg(p)\) be the degree (valency) of the vertex \(p\) in the graph \(\Gamma(G)\). According to \textit{A. R. Moghaddamfar, A. R. Zokayi} and \textit{M. R Darafsheh} [Algebra Colloq. 12, No. 3, 431-442 (2005; Zbl 1072.20015)], we define \(D(G)=(\deg(p_1),\dots,\deg(p_k))\), which is called the degree pattern of \(G\). The group \(G\) is called \(k\)-fold OD-characterizable if exist exactly \(k\) non-isomorphic finite groups with the same order and the same pattern as \(G\). Usually, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In the present paper, the authors first prove that each simple group from the pair \(\{B_n(q),C_n(q)\}\) is 2-fold OD-characterizable in the following cases: \(n=2^m>2\), \(q\) is odd and \(|\pi((q^n+1)/2)|=1\); \(n\) is an odd prime, \(q=3\) and \(|\pi((3^n-1)/2)|=1\); \(n=3\) and \(q=5\). They also prove that groups \(B_n(q)\) and \(C_n(q)\) are OD-characterizable in the following cases: \(n=2\) and \(|\pi((q^2+1)/(2,q-1)|=1\); \(n=2^m\geq 2\), \(q\) is even and \(|\pi((q^n+1))|=1\).
scientific article; zbMATH DE number 6000386