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Affine distance-transitive graphs with quadratic forms

Published online by Cambridge University Press:  24 October 2008

John Van Bon
John Van Bon
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, U.S.A.
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Extract

The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 507 - 517
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Van Bon, J.. Affine distance-transitive groups. Thesis, Rijksuniversiteit Utrecht (1990).Google Scholar
[2]Brouwer, A. E., Cohen, A. M. and Neumaier, A.. Distance-Regular Graphs. Ergebnisse der Math. und ihrer Grenzgebiete 3. Folge Band 18 (Springer-Verlag, 1989).Google Scholar
[3]Cohen, A. M. and Shult, E. E.. Affine polar spaces. Geom. Dedicata 35 (1990), 43–76.Google Scholar
[4]Cuypers, H.. Finite locally generalized quadrangles with affine planes. Preprint (1990).Google Scholar
[5]Cuypers, H. and Pasini, A.. Locally polar geometries with affine planes. Preprint (1990).Google Scholar
[6]Hall, J. I.. Graphs, geometry, 3-transpositions, and symplectic F2-transvection groups. Proc. London Math. Soc. (3) 58 (1989), 89–111.Google Scholar
[7]Liebeck, M. W.. The affine permutation groups of rank three. Proc. London Math. Soc. (3) 54 (1987), 447–516.Google Scholar