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 recogniz...

Given a matrix group G = 〈X〉 GL(d,q), specified by a set X of generators, it seems that a full structural exploration of G is necessary in order to answer even the simplest questions concerning G , such as finding |G| or testing the membership of any given matrix in G (cf. [LG,BB]). Currently, the standard approach to such an exploration is to set up a recursive scheme of homomorphisms, breakin...

The spectrum of a finite group is the set element orders this group. main goal paper to survey results concerning recognition simple groups by spectrum, in particular, list all for which problem solved.