Dempwolff group

Template:Short description 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 $2^{5.}{\mathrm{G}\mathrm{L}}_5({𝔽}_2)$ of ${\mathrm{G}\mathrm{L}}_5({𝔽}_2)$ by its natural module of order $2^5$. 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 $E_8$ 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.

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

References

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External links

• Dempwolff group at the atlas of groups.

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