On recognition of the projective special linear groups over binary fields

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. Let $L$ be the projective special linear group $L_n(2)$ with $n\ge3$. First, for all $n\ge3$ we establish that every finite group $G$ with $\omega(G)=\omega(L)$ has a unique non-abelian composition factor and this factor is isomorphic to $L$. Second, for some special series of integers $n$ we prove that $L$ is recognizable by spectrum, i. e. every finite group $G$ with $\omega(G)=\omega(L)$ is isomorphic to $L$.