In mathematical finite group theory, the Dempwolff group is a finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 of order 319979520 = 2¹⁵·3²·5·7·31, that is the unique nonsplit extension 2⁵·GL₅(F₂) of GL₅(F₂) by its natural module of order 2⁵. 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 E₈ as the subgroup fixing a certain lattice in the Lie algebra of E₈, and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

showed that that any extension of GL_n(F_q) by its natural module F 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 2³·GL₃(F₂) is a maximal subgroup of the Chevalley group G₂(F₃).
• The nonsplit extension 2⁴·GL₄(F₂) is a maximal subgroup of the sporadic Conway group
  Conway group
  In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
  
   Co₃.
• The nonsplit extension 2⁵·GL₅(F₂) is a maximal subgroup of the Thompson sporadic group Th.

External links

• Dempwolff group at the atlas of groups.