On \(r\)-recognition by prime graph of \(B_p(3)\) where \(p\) is an odd prime. (Q420581)

Let \(G\) be a finite group. The set of all primes dividing \(|G|\) is denoted by \(\pi(G)\). The prime graph \(\Gamma(G)\) of \(G\) is defined as follows: the vertices are the elements of \(\pi(G)\), and two distinct vertices \(p,q\) are joined by an edge if and only if there is an element of order \(pq\) in \(G\). In the present paper, the authors prove that if \(\Gamma(G)=\Gamma(B_p(3))\) where \(p\) is an odd prime, then one of the following holds: (a) \(p>3\) and \(G\cong B_p(3)\) or \(C_p(3)\); (b) \(p=3\) and \(G\cong B_3(3)\), \(C_3(3)\) or \(D_4(3)\), or \(G/O_2(G)\cong\Aut(^2B_2(8))\). -- This result generalizes the result of \textit{R. Shen, W. Shi} and \textit{M. R. Zinov'eva} [Sib. Math. J. 51, No. 2, 244-254 (2010); translation from Sib. Mat. Zh. 51, No. 2, 303-315 (2010; Zbl 1210.20020)] about the recognizability by element orders of the groups considered.