Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T22:19:56.359Z Has data issue: false hasContentIssue false

Involutions Associated with the Burkhardt Configuration in [4]

Published online by Cambridge University Press:  20 November 2018

A. F. Horadam*
Affiliation:
The University of New England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Horadam (11) has established the existence of a locus L in [8] (projective 8-space) having order 45 and dimension 4, which is invariant under a group of order 51840 X 81 (the Clifford similarity transform group CT). Associated with CT are two other groups, the Clifford collineation group CG of order 81, and the Clifford substitution group CS of order 51840. Furthermore, CS may be regarded as either a subgroup of CT, or a symplectic group of index matrices of size 4. Among the matrices of size 9 which perform the operations of CT, there is a set of 81 involutory, symmetric, orthogonal matrices JW. As collineation matrices in [8], these produce 81 pairs of invariant spaces Σ, Π of dimensions 3 and 4 respectively. These [4]'s give rise to a configuration C invariant under the operations of CT, consisting of 360 points, 1080 lines, 120 Jacobian planes, and 81 [4]'s, and their various inter-relationships.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Baker, H. F., A locus with 25920 linear self-transformations, Cambridge Tracts, 39 (1946).Google Scholar
2. Brahana, H. R., Pairs of generators of the known simple groups whose orders are less than one million, Ann. Math., 31 (1930), 529–49.Google Scholar
3. Dickson, L. E., Linear Groups with an Exposition of the Galois Field Theory (Leipzig, 1901).Google Scholar
4. Dieudonné, J., Les isomorphismes exceptionnels entre les groupes classiques finis, Can. J. Math., 6 (1954), 305–15.Google Scholar
5. Dieudonné, J., On the automorphisms of the classical groups, Mem. Amer. Math. Soc, 2 (1951).Google Scholar
6. Edge, W. L., Line geometry in three dimensions over GF(3) and the allied geometry of quadrics in four and five dimensions, Proc. Roy. Soc. A, 228 (1955), 129–46.Google Scholar
7. Edge, W. L., The conjugate classes of the cubic surface group in an orthogonal representation, Proc. Roy. Soc. A, 233 (1955), 126–46.Google Scholar
8. Edge, W. L., The characters of the cubic surface group, Proc. Roy. Soc. A, 237 (1956), 132-47.Google Scholar
9. Frame, J. S., The simple group of order 25920, Duke Math. J., 2 (1936), 477–84.Google Scholar
10. Frame, J. S. , The classes and representations of the groups of 27 lines and 28 bitangents, Ann. di Mat. pura ed applicata, 32 (1951), 83119.Google Scholar
11. Horadam, A. F., A locus in [8] invariant under a group of order 51840 X81, Quart. J. Math. (Oxford) (2) 8 (1957) 241-59.Google Scholar
12. Todd, J. A., On the simple group of order 25920, Proc. Roy. Soc. A, 189 (1947), 326–58.Google Scholar