Non-abelian representations of the slim dense near hexagons on 81 and 243 points
B. De Bruyn1
, B.K. Sahoo2
and N.S.N. Sastry2
1Department of Mathematics, Ghent University, Krijglaan 281 (S22), 9000 Gent, Belgium
2Statistics and Mathematics Unit, Indian Statistical Institute, Mysore Road, R. V. College Post, Bangalore 560059, India
2Statistics and Mathematics Unit, Indian Statistical Institute, Mysore Road, R. V. College Post, Bangalore 560059, India
DOI: 10.1007/s10801-010-0237-5
Abstract
We prove that the near hexagon Q(5,2)\times \mathbb L 3 Q(5,2)\times\mathbb{L}_{3} has a non-abelian representation in the extra-special 2-group 2 1+12 + 2^{1+12}_{+} and that the near hexagon Q(5,2) \otimes Q(5,2) has a non-abelian representation in the extra-special 2-group 2 1+18 - 2^{1+18}_{-}. The description of the non-abelian representation of Q(5,2) \otimes Q(5,2) makes use of a new combinatorial construction of this near hexagon.
Pages: 127–140
Keywords: keywords near hexagon; non-abelian representation; extra-special 2-group
Full Text: PDF
References
1. Brouwer, A.E., Cohen, A.M., Hall, J.I., Wilbrink, H.A.: Near polygons and Fischer spaces. Geom. Dedic. 49, 349-368 (1994)
2. De Bruyn, B.: Generalized quadrangles with a spread of symmetry. Eur. J. Comb. 20, 759-771 (1999)
3. De Bruyn, B.: On the number of nonisomorphic glued near hexagons. Bull. Belg. Math. Soc. Simon Stevin 7, 493-510 (2000)
4. De Bruyn, B., Thas, K.: Generalized quadrangles with a spread of symmetry and near polygons. Ill. J. Math. 46, 797-818 (2002)
5. Doerk, K., Hawkes, T.: Finite Soluble Groups. de Gruyter Expositions in Mathematics, vol.
4. Walter de Gruyter, Berlin (1992)
6. Gorenstein, D.: Finite Groups. Chelsea, New York (1980)
7. Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Oxford Mathematical Monographs. Oxford Science Publications. Clarendon, New York (1991)
8. Ivanov, A.A.: Non-abelian representations of geometries. In: “Groups and Combinatorics”-In Memory of Michio Suzuki. Adv. Stud. Pure Math., vol. 32, pp. 301-314. Math. Soc. Japan, Tokyo (2001)
9. Ivanov, A.A., Pasechnik, D.V., Shpectorov, S.V.: Non-abelian representations of some sporadic geometries. J. Algebra 181, 523-557 (1996)
10. Patra, K.L., Sahoo, B.K.: A non-abelian representation of the dual polar space DQ(2n, 2). Innov. Incidence Geom. 9, 177-188 (2009) J Algebr Comb (2011) 33: 127-140
11. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles, 2nd edn. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2009)
12. Sahoo, B.K., Sastry, N.S.N.: A characterization of finite symplectic polar spaces of odd prime order.
2. De Bruyn, B.: Generalized quadrangles with a spread of symmetry. Eur. J. Comb. 20, 759-771 (1999)
3. De Bruyn, B.: On the number of nonisomorphic glued near hexagons. Bull. Belg. Math. Soc. Simon Stevin 7, 493-510 (2000)
4. De Bruyn, B., Thas, K.: Generalized quadrangles with a spread of symmetry and near polygons. Ill. J. Math. 46, 797-818 (2002)
5. Doerk, K., Hawkes, T.: Finite Soluble Groups. de Gruyter Expositions in Mathematics, vol.
4. Walter de Gruyter, Berlin (1992)
6. Gorenstein, D.: Finite Groups. Chelsea, New York (1980)
7. Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Oxford Mathematical Monographs. Oxford Science Publications. Clarendon, New York (1991)
8. Ivanov, A.A.: Non-abelian representations of geometries. In: “Groups and Combinatorics”-In Memory of Michio Suzuki. Adv. Stud. Pure Math., vol. 32, pp. 301-314. Math. Soc. Japan, Tokyo (2001)
9. Ivanov, A.A., Pasechnik, D.V., Shpectorov, S.V.: Non-abelian representations of some sporadic geometries. J. Algebra 181, 523-557 (1996)
10. Patra, K.L., Sahoo, B.K.: A non-abelian representation of the dual polar space DQ(2n, 2). Innov. Incidence Geom. 9, 177-188 (2009) J Algebr Comb (2011) 33: 127-140
11. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles, 2nd edn. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2009)
12. Sahoo, B.K., Sastry, N.S.N.: A characterization of finite symplectic polar spaces of odd prime order.
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