In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite fields of odd order, depending on a certain congruence, or to the Mathieu group . proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in , Ch. 7), and presented in some detail in .