# Alperin-Brauer-Gorenstein-Theorem

In Mathematik, 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
{Anzeigestil M_{11}}
. Alperin, Brauer & Gorenstein (1970) proved this in the course of 261 Seiten. 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

Im Mathematik, das Alperin-Brauer-Gorenstein-Theorem characterizes the finite simple groups mit quasidihedral or wreathed[1] Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups oder projective special unitary groups over a finite field of odd order, depending on a certain congruence, or to the Mathieu group

$"{displaystyle$

.

Alperin, Brauer & Gorenstein (1970) proved this in the course of 261 Seiten.

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).

Índice

## Anmerkungen[]

1. ^ A 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direktes Produkt of two cyclic groups of the same order, das ist, if it is the wreath product of a cyclic 2-group with the symmetric group an 2 Punkte.

## Verweise[]

• Alperin, J. L.Brauer, R.Gorenstein, D. (1970), "Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.", Transaktionen der American Mathematical Society, Amerikanische Mathematische Gesellschaft, 151 (1): 1–261, doi:10.2307/1995627, ISSN 0002-9947, JSTOR 1995627, HERR 0284499

• Gorenstein, D. (1968), Finite groups, Harper & Row Publishers, HERR 0231903
• Kwon, T.; Lee, K.; Cho, ICH.; Park, S. (1980), "On finite groups with quasidihedral Sylow 2-groups", Journal of the Korean Mathematical Society, 17 (1): 91–97, ISSN 0304-9914, HERR 0593804, archived from the original an 2011-07-22, abgerufen 2010-07-16

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