Dempwolff group

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 $2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})$ of $\mathrm{GL}_{5}(\mathbb{F}_{2})$ by its natural module of order $2^5$ . 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 $E_{8}$ as the subgroup fixing a certain lattice in the Lie algebra of $E_{8}$ , 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 any extension of $\mathrm{GL}_{n}(\mathbb{F}_{q})$ by its natural module $\mathbb{F}_{q}^{n}$ splits if $q>2$ . Note that this theorem does not necessarily apply to extensions of $\mathrm{SL}_{n}(\mathbb{F}_{q})$ ; for example, there is a non-split extension $5^{3\,.}\mathrm{SL}_{n}(\mathbb{F}_{q})$ , which is a maximal subgroup of the Lyons group. {{harvtxt|Dempwolff|1973}} showed that it also splits if $n$ 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 $2^{3\,.}\mathrm{GL}_{3}(\mathbb{F}_{2})$ is a maximal subgroup of the Chevalley group $G_{2}(\mathbb{F}_{3})$ .
The nonsplit extension $2^{4\,.}\mathrm{GL}_{4}(\mathbb{F}_{2})$ is a maximal subgroup of the sporadic Conway group Co₃.
The nonsplit extension $2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})$ is a maximal subgroup of the Thompson sporadic group Th.

References

{{Citation | last1=Dempwolff | first1=Ulrich | title=On extensions of an elementary abelian group of order 2⁵ 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| doi-access=free }}
{{Citation | last1=Griess | first1=Robert L. | title=On a subgroup of order 2¹⁵ . ¦GL(5,2)¦ in E₈(C), the Dempwolff group and Aut(D₈°D₈°D₈) | 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| url=https://deepblue.lib.umich.edu/bitstream/2027.42/21778/1/0000172.pdf | hdl=2027.42/21778 | hdl-access=free }}
{{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 E₈(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 E₈ | 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| doi-access=free }}

External links

[http://brauer.maths.qmul.ac.uk/Atlas/v3/misc/25L52/ Dempwolff group] at the atlas of groups.

Category:Finite groups