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 of by its natural module of order . 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 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.
Huppert (1967, p.124) showed that any extension of by its natural module splits if , and Dempwolff (1973) 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.
- 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, MR 0393276
- Dempwolff, Ulrich (1973), "On the second cohomology of GL(n,2)", Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics, 16: 207–209, ISSN 0263-6115, MR 0357639, doi:10.1017/S1446788700014221
- 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, ISSN 0021-8693, MR 0407149, doi:10.1016/0021-8693(76)90097-1
- Huppert, Bertram (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, 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, ISSN 0024-6093, MR 0409630, doi:10.1112/blms/8.2.161
- Thompson, John G. (1976), "A conjugacy theorem for E8", Journal of Algebra, 38 (2): 525–530, ISSN 0021-8693, MR 0399193, doi:10.1016/0021-8693(76)90235-0
- Dempwolff group at the atlas of groups.