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 $2^\mathrm_(\mathbb_)$ of $\mathrm_(\mathbb_)$ by its natural module of order $2^5$. The uniqueness of such a nonsplit extension was shown by , and the existence by , who showed using some computer calculations of that the Dempwolff group is contained in the compact Lie group $E_$ as the subgroup fixing a certain lattice in the Lie algebra of $E_$, and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

showed that any extension of $\mathrm_(\mathbb_)$ by its natural module $\mathbb_^$ splits if $q>2$, and 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 $2^\mathrm_(\mathbb_)$ is a maximal subgroup of the Chevalley group $G_(\mathbb_)$.
*The nonsplit extension $2^\mathrm_(\mathbb_)$ is a maximal subgroup of the sporadic Conway group Co₃.
*The nonsplit extension $2^\mathrm_(\mathbb_)$ is a maximal subgroup of the Thompson sporadic group Th.

References

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External links

Dempwolff group
at the atlas of groups.
{{algebra-stub

Finite groups