Recognition by element orders for simple linear and unitary groups

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $\omega(H)=\omega(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.