Polarized non-abelian representations of slim near-polar spaces. (English) Zbl 1342.05020
Summary: E. E. Shult [Bull. Belg. Math. Soc. - Simon Stevin 4, No. 2, 299–316 (1997; Zbl 0923.51013)], introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in [B. K. Sahoo and N. S. Narasimha Sastry, J. Algebr. Comb. 29, No. 2, 195–213 (2009; Zbl 1226.05269)] for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding.
MSC:
| 05B25 | Combinatorial aspects of finite geometries |
| 51A45 | Incidence structures embeddable into projective geometries |
| 51A50 | Polar geometry, symplectic spaces, orthogonal spaces |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
near-polar space; universal non-abelian representation; polarized non-abelian representation; universal projective embedding; minimal polarized embedding; extraspecial 2-group; combinatorial group theoryReferences:
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