# Alperin–Brauer–Gorenstein theorem

In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed[1] 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

{displaystyle 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 et al. (1980).

# Alperin–Brauer–Gorenstein theorem

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In mathematics, the **Alperin–Brauer–Gorenstein theorem** characterizes the finite simple groups with quasidihedral or wreathed^{[1]} 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

.

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 et al. (1980).

## Notes[]

**^**A 2-group is**wreathed**if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic groups of the same order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points.

## References[]

Alperin, J. L.Brauer, R.Gorenstein, D. (1970), "Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.",

*Transactions of the American Mathematical Society*, American Mathematical Society,**151**(1): 1–261, doi:10.2307/1995627, ISSN 0002-9947, JSTOR 1995627, MR 0284499- Gorenstein, D. (1968),
*Finite groups*, Harper & Row Publishers, MR 0231903 - Kwon, T.; Lee, K.; Cho, I.; Park, S. (1980), "On finite groups with quasidihedral Sylow 2-groups",
*Journal of the Korean Mathematical Society*,**17**(1): 91–97, ISSN 0304-9914, MR 0593804, archived from the original on 2011-07-22, retrieved 2010-07-16

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