Recognizability of the Groups $$PSp_8(7^m)$$ by the Set of Element Orders

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.