On the order of a non-abelian representation group of a slim dense near hexagon. (English) Zbl 1226.05269
Summary: 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 \(|R|=2^{\beta}, 1+NPdim(S)\leq \beta \leq 1+dimV(S)\), 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 \(\beta =1+NPdim(S)\), 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.
MSC:
| 05E99 | Algebraic combinatorics |
| 05B25 | Combinatorial aspects of finite geometries |
| 51E12 | Generalized quadrangles and generalized polygons in finite geometry |
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