## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 11, Number 3 (2004), 321-341.

### Near polygons having a big sub near polygon isomorphic to $\mathbb G_n$

#### Abstract

In [7] a new infinite class $\mathbb G_n$, $n \geq 0$, of near polygons was defined. The near $2n$-gon $\mathbb G_n$ has the property that it contains $\mathbb G_{n-1}$ as a big geodetically closed sub near polygon. In this paper, we determine all near $2n$-gons, $n \geq 4$, having $\mathbb G_{n-1}$ as a big geodetically closed sub near $2(n-1)$-gon under the additional assumption that every two points at distance 2 have at least two common neighbours. We will prove that such a near $2n$-gon is isomorphic to either $\mathbb G_n$, $\mathbb G_{n-1} \otimes \mathbb G_2$, or $\mathbb G_{n-1} \times L$ for some line $L$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 11, Number 3 (2004), 321-341.

**Dates**

First available in Project Euclid: 24 August 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1093351376

**Digital Object Identifier**

doi:10.36045/bbms/1093351376

**Mathematical Reviews number (MathSciNet)**

MR2098411

**Zentralblatt MATH identifier**

1067.05016

**Subjects**

Primary: 05B20: Matrices (incidence, Hadamard, etc.) 51E12: Generalized quadrangles, generalized polygons 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

**Keywords**

near polygon generalized quadrangle hermitean variety

#### Citation

De Bruyn, Bart. Near polygons having a big sub near polygon isomorphic to $\mathbb G_n$. Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 3, 321--341. doi:10.36045/bbms/1093351376. https://projecteuclid.org/euclid.bbms/1093351376