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 \( 2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2}) \) of \( \mathrm{GL}_{5}(\mathbb{F}_{2}) \) by its natural module of order \( 2^5 \). 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 \( E_{8} \) 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.

Huppert (1967, p.124) showed that that any extension of \mathrm{GL}_{n}(\mathbb{F}_{q}) by its natural module \mathbb{F}_{q}^{n} splits if q>2, and Dempwolff (1973) 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{GL}_{3}(\mathbb{F}_{2}) is a maximal subgroup of the Chevalley group G_{2}(\mathbb{F}_{3}).
The nonsplit extension 2^{4\,.}\mathrm{GL}_{4}(\mathbb{F}_{2}) is a maximal subgroup of the sporadic Conway group Co3.
The nonsplit extension 2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2}) is a maximal subgroup of the Thompson sporadic group Th.

References

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, doi:10.1017/S1446788700014221, ISSN 0263-6115, MR 0357639
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, doi:10.1016/0021-8693(76)90097-1, ISSN 0021-8693, MR 0407149
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, doi:10.1112/blms/8.2.161, ISSN 0024-6093, MR 0409630
Thompson, John G. (1976), "A conjugacy theorem for E8", Journal of Algebra 38 (2): 525–530, doi:10.1016/0021-8693(76)90235-0, ISSN 0021-8693, MR 0399193

External links

Dempwolff group at the atlas of groups.

Mathematics Encyclopedia