Dempwolff group

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 2¹⁵·3²Ã‚·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 , and the existence by , who showed using some computer calculations of 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.

showed that any extension of by its natural module splits if . Note that this theorem does not necessarily apply to extensions of ; for example, there is a non-split extension , which is a maximal subgroup of the Lyons group. 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:

• The nonsplit extension is a maximal subgroup of the Chevalley group .
• The nonsplit extension is a maximal subgroup of the sporadic Conway group Co₃.
• The nonsplit extension is a maximal subgroup of the Thompson sporadic group Th.

References

External links

• Dempwolff group at the atlas of groups.