The homotopy type of BG for some small matrix groups G

Let q be a power of an odd prime. We prove that the mod-2 cohomologies 
of BGL_2(F_q), BSL_3(F_q), and BGL_3(F_q), as algebras over the mod-2 
Steenrod algebra, together with the associated Bockstein spectral sequence, 
determine the homotopy types of respectively BGL_2(F_q), BSL_3(F_q), and GL_3(F_q).