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An alternative construction of the G2(2)-hexagon

Koichi Inoue

TL;DR

The paper presents an explicit, alternative construction of the $G_2(2)$-hexagon starting from a ${U}_3(2)$-geometry, providing an elementary route to a split Cayley hexagon of order $2$ and a second, non-equivalent degree-$63$ permutation representation of ${G_2(2)}$. It develops a ${U}_3(2)$-based design on a 6-dimensional ${ m GF}(2)$ space via a Hermitian form over ${ m GF}(4)$, showing the resulting incidence structure is a generalized hexagon whose automorphism group is ${G_2(2)}$. The construction leverages a total-isotropic line framework, connectivity of the concurrency graph, and the Cuypers–Steinbach theorem to establish isomorphism with the known ${G_2(2)}$-hexagon. This provides a new, more elementary viewpoint and broadens the toolkit for exploring ${G_2(2)}$ and its permutation representations.

Abstract

In this note, we give an alternative and explicit construction of the $G_2(2)$-hexagon from a $U_3(2)$-geometry.

An alternative construction of the G2(2)-hexagon

TL;DR

The paper presents an explicit, alternative construction of the -hexagon starting from a -geometry, providing an elementary route to a split Cayley hexagon of order and a second, non-equivalent degree- permutation representation of . It develops a -based design on a 6-dimensional space via a Hermitian form over , showing the resulting incidence structure is a generalized hexagon whose automorphism group is . The construction leverages a total-isotropic line framework, connectivity of the concurrency graph, and the Cuypers–Steinbach theorem to establish isomorphism with the known -hexagon. This provides a new, more elementary viewpoint and broadens the toolkit for exploring and its permutation representations.

Abstract

In this note, we give an alternative and explicit construction of the -hexagon from a -geometry.
Paper Structure (6 sections, 6 theorems, 1 equation)

This paper contains 6 sections, 6 theorems, 1 equation.

Table of Contents

  1. Abstract.
  2. Keywords:
  3. Introduction
  4. Preliminaries
  5. The $G_2(2)$-hexagon
  6. The Construction

Key Result

Lemma 4.2

$\bm{S}$ is a partial linear space of order $(2,2)$.

Theorems & Definitions (11)

  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • Theorem 4.6
  • ...and 1 more