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 $M_{11}$ . 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

^ 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.

Alperin, J. L.; Brauer, R.; Gorenstein, D. (1970), "Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.", Transactions of the American Mathematical Society, 151 (1), American Mathematical Society: 1–261, doi:10.2307/1995627, ISSN 0002-9947, JSTOR 1995627, MR 02844991-261&rft.date=1970&rft.issn=0002-9947&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0284499#id-name=MR&rft_id=https://www.jstor.org/stable/1995627#id-name=JSTOR&rft_id=info:doi/10.2307/1995627&rft.aulast=Alperin&rft.aufirst=J. L.&rft.au=Brauer, R.&rft.au=Gorenstein, D.&rfr_id=info:sid/en.wikipedia.org:Alperin–Brauer–Gorenstein theorem" class="Z3988">
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 2010-07-1691-97&rft.date=1980&rft.issn=0304-9914&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=593804#id-name=MR&rft.aulast=Kwon&rft.aufirst=T.&rft.au=Lee, K.&rft.au=Cho, I.&rft.au=Park, S.&rft_id=http://kms.or.kr/home/journal/include/downloadPdfJournal.asp?articleuid=%7B71EE4232%2D6997%2D4030%2D8CA7%2D85CDBCB5A2CC%7D&rfr_id=info:sid/en.wikipedia.org:Alperin–Brauer–Gorenstein theorem" class="Z3988">

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