Abstract
The set of orders of the elements of a finite group \(G\) is called the spectrum of \(G\) . A group \(G\) is said to be unrecognizable by spectrum if there are infinitely many pairwise nonisomorphic finite groups that have the same spectrum as \(G\) . There is a conjecture that every finite simple classical group unrecognizable by spectrum is contained in the following list: \(PSL_3(3)\) , \(PSU_3(q)\) , \(PSU_5(2)\) , \(PSp_4(q)\) , \(PSp_8(q)\) , and \(P\Omega_9(q)\) . The only groups in this list that were not known to be unrecognizable by spectrum are \(PSp_8(7^m)\) . In the present paper, it is shown that \(PSp_8(7^m)\) are not unrecognizable and, moreover, any of these groups is uniquely (up to isomorphism) determined by its spectrum in the class of all finite groups.
