# 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 fields of odd order, depending on a certain congruence, or to the Mathieu group ${\displaystyle M_{11}}$. Template:Harvtxt proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in Template:Harvtxt, and presented in some detail in Template:Harvtxt.

## Notes

1. 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

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