On the order of a non-abelian representation group of a slim dense near hexagon
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|=2β, 1+NPdim(S)≤β≤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.
KeywordsNear polygons Non-abelian representations Generalized quadrangles Extraspecial 2-groups
- 3.Cameron, P.J.: Projective and polar spaces. Available from http://www.maths.qmul.ac.uk/~pjc/
- 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 Google Scholar