Dempwolff group

Short description: Finite 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 $2^{5.}{\mathrm{G}\mathrm{L}}_5({\mathbb{𝔽}}_2)$ of ${\mathrm{G}\mathrm{L}}_5({\mathbb{𝔽}}_2)$ by its natural module of order $2^5$. The uniqueness of such a nonsplit extension was shown by (Dempwolff 1972), and the existence by (Thompson 1976), who showed using some computer calculations of (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.

(Huppert 1967) showed that any extension of ${\mathrm{G}\mathrm{L}}_n({\mathbb{𝔽}}_q)$ by its natural module ${\mathbb{𝔽}}_q^n$ splits if $q>2$. Note that this theorem does not necessarily apply to extensions of ${\mathrm{S}\mathrm{L}}_n({\mathbb{𝔽}}_q)$; for example, there is a non-split extension $5^{3.}{\mathrm{S}\mathrm{L}}_3({\mathbb{𝔽}}_5)$, which is a maximal subgroup of the Lyons group. (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 $2^{3.}{\mathrm{G}\mathrm{L}}_3({\mathbb{𝔽}}_2)$ is a maximal subgroup of the Chevalley group $G_2({\mathbb{𝔽}}_3)$.
• The nonsplit extension $2^{4.}{\mathrm{G}\mathrm{L}}_4({\mathbb{𝔽}}_2)$ is a maximal subgroup of the sporadic Conway group Co₃.
• The nonsplit extension $2^{5.}{\mathrm{G}\mathrm{L}}_5({\mathbb{𝔽}}_2)$ is a maximal subgroup of the Thompson sporadic group Th.

References

• Dempwolff, Ulrich (1972), "On extensions of an elementary abelian group of order 2⁵ by GL(5,2)", Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova 48: 359–364, ISSN 0041-8994, http://www.numdam.org/item?id=RSMUP_1972__48__359_0 
• Dempwolff, Ulrich (1973), "On the second cohomology of GL(n,2)", Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics 16: 207–209, doi:10.1017/S1446788700014221, ISSN 0263-6115 
• Griess, Robert L. (1976), "On a subgroup of order 2¹⁵ . ¦GL(5,2)¦ in E₈(C), the Dempwolff group and Aut(D₈°D₈°D₈)", Journal of Algebra 40 (1): 271–279, doi:10.1016/0021-8693(76)90097-1, ISSN 0021-8693, https://deepblue.lib.umich.edu/bitstream/2027.42/21778/1/0000172.pdf 
• Huppert, Bertram (1967) (in German), Endliche Gruppen, Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, OCLC 527050 
• Smith, P. E. (1976), "A simple subgroup of M? and E₈(3)", The Bulletin of the London Mathematical Society 8 (2): 161–165, doi:10.1112/blms/8.2.161, ISSN 0024-6093 
• Thompson, John G. (1976), "A conjugacy theorem for E₈", Journal of Algebra 38 (2): 525–530, doi:10.1016/0021-8693(76)90235-0, ISSN 0021-8693 

External links

• Dempwolff group at the atlas of groups.