Alperin–Brauer–Gorenstein theorem

Statement

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 field of odd order, depending on a certain congruence, or to the Mathieu group $M_{11}$. (Alperin Brauer Gorenstein 1970) proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in (Gorenstein 1968, Ch. 7), and presented in some detail in (Kwon Lee Cho Park 1980).