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  "body": "\n## Statement\n\nIn 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|0ZXZ82FTX2]] 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).\n",
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