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 [math]\displaystyle{ M_{11} }[/math]. (Alperin Brauer) proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in (Gorenstein 1968), and presented in some detail in (Kwon Lee).
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
- Gorenstein, D. (1968), Finite groups, Harper & Row Publishers
- 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, http://kms.or.kr/home/journal/include/downloadPdfJournal.asp?articleuid=%7B71EE4232%2D6997%2D4030%2D8CA7%2D85CDBCB5A2CC%7D, retrieved 2010-07-16
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