Almost recognizability by spectrum of finite simple linear groups of prime dimension. (Q1760539)

Let \(G\) be a finite group. The spectrum of \(G\) is the set of element orders of \(G\) and is denoted by \(w(G)\). The recognition problem for \(G\) is to determine the number of isomorphism classes of finite groups \(H\) with \(w(H)=w(G)\). The main result of the paper under review is that if \(L=\text{PSL}(n,q)\) where \(n\) is a prime greater than 3 and if \(G\) is finite group with \(w(L)=w(G)\), then \(L<G<\Aut\,L\). This implies that the number of isomorphism classes of the group \(G\) is finite, i.e. the group \(L\) is almost recognizable by the set of element orders.