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.
Keywords:orders of elements, simple group, symplectic group, orthogonal group,
representations in defining characteristic.