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 $M_{11}$. 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).

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

• 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 593804