On the order of a non-abelian representation group of a slim dense near hexagon
Binod Kumar Sahoo1
and N.S.Narasimha Sastry2
1National Institute of Technology Department of Mathematics Rourkela 769008 India
2Indian Statistical Institute Statistics and Mathematics Unit 8th Mile, Mysore Road, R.V. College Post Bangalore 560059 India
2Indian Statistical Institute Statistics and Mathematics Unit 8th Mile, Mysore Road, R.V. College Post Bangalore 560059 India
DOI: 10.1007/s10801-008-0129-0
Abstract
In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and | R|=2 β , 1+ NPdim( S)\leq β \leq 1+ dimV( S), where NPdim( S) is the near polygon embedding dimension of S and dimV( S) is the dimension of the universal representation module V( S) of S. Further, if β =1+ NPdim( S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4.
Pages: 195–213
Keywords: keywords near polygons; non-abelian representations; generalized quadrangles; extraspecial 2-groups
Full Text: PDF
References
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2. Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedicata 14(2), 145-176 (1983)
3. Cameron, P.J.: Projective and polar spaces. Available from
4. De Bruyn, B.: Near Polygons. Frontiers in Mathematics. Birkhäuser, Basel (2006)
5. De Bruyn, B., Vandecasteele, P.: Near polygons with a nice chain of sub-near polygons. J. Combin. Theory Ser. A 108(2), 297-311 (2004)
6. De Bruyn, B., Vandecasteele, P.: The classification of the slim dense near octagons. European J. Combin. 28(1), 410-428 (2007)
7. Doerk, K., Hawkes, T.: Finite Soluble Groups. de Gruyter Expositions in Mathematics, vol.
4. Walter de Gruyter & Co., Berlin (1992)
8. Ivanov, A.A.: Non-abelian representations of geometries. Groups and combinatorics-in memory of Michio Suzuki. Adv. Stud. Pure Math. 32, 301-314 (2001) Math. Soc. Japan, Tokyo
9. Ivanov, A.A., Pasechnik, D.V., Shpectorov, S.V.: Non-abelian representations of some sporadic geometries. J. Algebra 181(2), 523-557 (1996)
10. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics, vol.
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11. Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. European J. Combin. 8(2), 179- 185 (1987)
12. Sahoo, B.K., Sastry, N.S.N.: A characterization of finite symplectic polar spaces of odd prime order.