On recognition of low-dimensional linear and unitary groups by spectrum

Finite groups are isospectral if they have the same sets of element orders. We complete the description of the finite groups isospectral to the simple groups $U_5(q)$, $L_6(q)$, and $U_6(q)$, with $q$ odd. We also prove that the exceptional groups of Lie type do not occur among the composition factors of finite groups isospectral to simple classical groups.