Alperin–Brauer–Gorenstein theorem

In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathedA 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. 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 $M_{11}$ . {{harvtxt|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 {{harvtxt|Gorenstein|1968|loc=Ch. 7}}, and presented in some detail in {{harvtxt|Kwon|Lee|Cho|Park|1980}}.

Notes

{{Citation | last1=Alperin | first1=J. L. | author1-link=J. L. Alperin | last2=Brauer | first2=R. | author2-link=Richard Brauer | last3=Gorenstein | first3=D. | author3-link=Daniel Gorenstein | title=Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups. | doi=10.2307/1995627 | mr=0284499 | year=1970 | journal=Transactions of the American Mathematical Society | publisher=American Mathematical Society | issn=0002-9947 | volume=151 | pages=1–261 | jstor=1995627 | issue=1}}
{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | publisher=Harper & Row Publishers | mr=0231903 | year=1968}}
{{Citation | last1=Kwon | first1=T. | last2=Lee | first2=K. | last3=Cho | first3=I. | last4=Park | first4=S. | title=On finite groups with quasidihedral Sylow 2-groups | url=http://kms.or.kr/home/journal/include/downloadPdfJournal.asp?articleuid=%7B71EE4232%2D6997%2D4030%2D8CA7%2D85CDBCB5A2CC%7D | mr=593804 | year=1980 | journal=Journal of the Korean Mathematical Society | issn=0304-9914 | volume=17 | issue=1 | pages=91–97 | access-date=2010-07-16 | archive-url=https://web.archive.org/web/20110722141219/http://kms.or.kr/home/journal/include/downloadPdfJournal.asp?articleuid=%7B71EE4232-6997-4030-8CA7-85CDBCB5A2CC%7D | archive-date=2011-07-22 | url-status=dead }}