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In mathematical finite group theory, the '''Dempwolff group''' is a [[finite group]] of order 319979520 =&nbsp;2<sup>15</sup>·3<sup>2</sup>·5·7·31, that is the unique nonsplit extension <math>2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})</math> of <math>\mathrm{GL}_{5}(\mathbb{F}_{2})</math> by its natural module of order <math>2^5</math>. The uniqueness of such a nonsplit extension was shown by {{harvtxt|Dempwolff|1972}}, and the existence by {{harvtxt|Thompson|1976}}, who showed using some computer calculations of {{harvtxt|Smith|1976}} that the Dempwolff group is contained in the compact Lie group <math>E_{8}</math> as the subgroup fixing a certain lattice  in the Lie algebra of <math>E_{8}</math>and is also contained in the  [[Thompson sporadic group]] (the full automorphism group of this lattice) as a maximal subgroup.
 
{{harvtxt|Huppert|1967|loc=p.124}} showed that that  any extension of <math>\mathrm{GL}_{n}(\mathbb{F}_{q})</math> by its natural module <math>\mathbb{F}_{q}^{n}</math> splits if <math>q>2</math>, and {{harvtxt|Dempwolff|1973}} showed that it also splits  if <math>n</math> is not 3, 4, or 5, and in each of these three cases there is just one non-split extension. These three nonsplit extensions can be constructed as follows:  
*The nonsplit extension <math>2^{3\,.}\mathrm{GL}_{3}(\mathbb{F}_{2})</math> is a maximal subgroup of the [[Chevalley group]] <math>G_{2}(\mathbb{F}_{3})</math>.
*The nonsplit extension <math>2^{4\,.}\mathrm{GL}_{4}(\mathbb{F}_{2})</math> is a maximal subgroup of the sporadic [[Conway group]]&nbsp;Co<sub>3</sub>.
*The nonsplit extension <math>2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})</math> is a maximal subgroup of the [[Thompson sporadic group]]&nbsp;Th.
 
==References==
*{{Citation | last1=Dempwolff | first1=Ulrich | title=On extensions of an elementary abelian group of order 2<sup>5</sup> by GL(5,2) | url=http://www.numdam.org/item?id=RSMUP_1972__48__359_0 | mr=0393276 | year=1972 | journal=Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova | issn=0041-8994 | volume=48 | pages=359–364}}
*{{Citation | last1=Dempwolff | first1=Ulrich | title=On the second cohomology of GL(n,2) | doi=10.1017/S1446788700014221 | mr=0357639 | year=1973 | journal=Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics | issn=0263-6115 | volume=16 | pages=207–209}}
*{{Citation | last1=Griess | first1=Robert L. | title=On a subgroup of order 2<sup>15</sup> . ¦GL(5,2)¦ in E<sub>8</sub>(C), the Dempwolff group and Aut(D<sub>8</sub>°D<sub>8</sub>°D<sub>8</sub>) | doi=10.1016/0021-8693(76)90097-1 | mr=0407149 | year=1976 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=40 | issue=1 | pages=271–279}}
*{{Citation | last1=Huppert | first1=Bertram | author1-link=Bertram Huppert | title=Endliche Gruppen | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=German | isbn=978-3-540-03825-2 | oclc=527050 | mr=0224703 | year=1967}}
*{{Citation | last1=Smith | first1=P. E. | title=A simple subgroup of M? and E<sub>8</sub>(3) | doi=10.1112/blms/8.2.161  | mr=0409630 | year=1976 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=8 | issue=2 | pages=161–165}}
*{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=A conjugacy theorem for E<sub>8</sub> | doi=10.1016/0021-8693(76)90235-0 | mr=0399193 | year=1976 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=38 | issue=2 | pages=525–530}}
 
==External links==
*[http://brauer.maths.qmul.ac.uk/Atlas/v3/misc/25L52/ Dempwolff group] at the atlas of groups.
 
[[Category:Finite groups]]

Revision as of 03:07, 14 January 2014

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Template:Harvtxt, and the existence by Template:Harvtxt, who showed using some computer calculations of Template:Harvtxt that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

Template:Harvtxt showed that that any extension of by its natural module splits if , and Template:Harvtxt showed that it also splits if is not 3, 4, or 5, and in each of these three cases there is just one non-split extension. These three nonsplit extensions can be constructed as follows:

References

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External links