8 Properties of element orders in covers for L n ( q ) and U n ( q ) Andrei

We show that if a finite simple group G isomorphic to PSLn(q) or PSUn(q), where either n 6= 4, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSLn(q), n 6= 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka notebook.