---
tag: A9CXS44AVE
kind: statement
role: proposition
name: Alperin–Brauer–Gorenstein theorem
sources:
  - wikipedia: en:8918642@r1282809150
    locator: Alperin–Brauer–Gorenstein theorem
    note: adopted
    mention: wikipedia://en/8918642
    sections: [Statement]
same_as:
  wikidata: Q4734844
  wikipedia: Alperin–Brauer–Gorenstein theorem
inventory: [EZCCHA3W66]
---

## 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|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).