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| In [[mathematics]], the '''Alperin–Brauer–Gorenstein theorem''' characterizes the finite [[simple group]]s with [[quasidihedral]] or wreathed<ref>A 2-group is '''wreathed''' if it is a nonabelian [[semidirect product]] of a [[maximal subgroup]] that is a [[direct product of groups|direct product]] of two [[cyclic group]]s of the same order, that is, if it is the [[wreath product]] of a cyclic 2-group with the [[symmetric group]] on 2 points.</ref> [[Sylow subgroup|Sylow 2-subgroups]]. These are isomorphic either to three-dimensional [[projective special linear group]]s or [[projective special unitary group]]s over a [[finite field]]s of odd order, depending on a certain congruence, or to the [[Mathieu group]] <math>M_{11}</math>. {{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}}.
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| ==Notes==
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| <references/> | |
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| ==References==
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| *{{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}}
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| *{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | publisher=[[Harper & Row Publishers]] | mr=0231903 | year=1968}}
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| *{{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}}
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| {{DEFAULTSORT:Alperin-Brauer-Gorenstein theorem}}
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| [[Category:Finite groups]]
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| [[Category:Theorems in algebra]]
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| {{Abstract-algebra-stub}}
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Latest revision as of 02:36, 15 November 2014
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