The $\mathrm {L}_3(4)$ near octagon

Abstract

In recent work we constructed two new near octagons, one related to the finite simple group $\mathrm {G}_2(4)$ and another one as a sub-near-octagon of the former. In the present paper, we give a direct construction of this sub-near-octagon using a split extension of the group $\mathrm {L}_3(4)$. We derive several geometric properties of this $\mathrm {L}_3(4)$ near octagon, and determine its full automorphism group. We also prove that the $\mathrm {L}_3(4)$ near octagon is closely related to the second subconstituent of the distance-regular graph on 486 vertices discovered by Soicher (Eur J Combin 14:501–505, 1993).

Appendix: The \(\mathrm {G}_2(4)\) near octagon and a distance-regular graph of Soicher

In this appendix we establish the connection between the near octagon ${\mathbb O}_1$ and the distance-regular graph $\Sigma$ discovered by Soicher in [10], which has intersection array $\{416, 315, 64, 1; 1, 32, 315, 416\}$. The graph $\Sigma$ is a triple cover of the Suzuki graph (in the sense of [10]) and has $\mathrm {Aut}(\Sigma ) \cong 3 \cdot \mathrm {Suz}{:}2$. Under the action of the stabilizer (which is isomorphic to $\mathrm {G}_2(4){:}2$) of its automorphism group with respect to a vertex x, the orbits are equal to $\Sigma _0(x)$, $\Sigma _1(x)$, $\Sigma _2(x)$, $\Sigma _3(x)$ and $\Sigma _4(x)$, with sizes 1, 416, 4095, 832 and 2, respectively (so in particular, the graph $\Sigma$ is distance-transitive).

Let x be a fixed vertex of $\Sigma$. The local graph $\Sigma _x$ is isomorphic to the well-known $\mathrm {G}_2(4)$-graph, which is a strongly regular graph with parameters $(v, k, \lambda , \mu ) = (416, 100, 36, 20)$. Let $G = \mathrm {Aut}(\Sigma )$. Then the stabilizer $G_x$ is isomorphic to the group $\mathrm {G}_2(4){:}2$, and it is the full automorphism group of the local graph $\Sigma _x$. Let ${\mathbb O}_1(x)$ denote the near octagon defined on the central involutions of $G_x \cong \mathrm {G}_2(4){:}2$ (in the sense of Proposition 1.1). We have computationally verified (see [3]) that every such involution $\sigma$ (which is a point of ${\mathbb O}_1(x)$) fixes 32 points of the $\mathrm {G}_2(4)$-graph $\Sigma _x$, which we denote by $X_\sigma$. Moreover, the elements of $X_\sigma$ determine the involution $\sigma$ uniquely, as the group generated by $\sigma$ is the unique subgroup of $\mathrm {G}_2(4){:}2$ which fixes $X_\sigma$ pointwise. Therefore, the 4095 points of ${\mathbb O}_1(x)$ are in bijective correspondence with 4095 such special 32-sets in the $\mathrm {G}_2(4)$-graph $\Sigma _x$.

In Soicher’s graph $\Sigma$ we can computationally check that for every vertex $y \in \Sigma _2(x)$, the set $\Sigma _1(x) \cap \Sigma _1(y)$ is a special 32-set of the $\mathrm {G}_2(4)$-graph $\Sigma _x$, i.e., there exists a unique central involution $\sigma _y$ of $\mathrm {G}_2(4){:}2$ which fixes this set $X_y$ pointwise. In this manner, we get a map $y \mapsto \sigma _y$ between the 4095 elements of $\Sigma _2(x)$ and the 4095 central involutions of $\mathrm {G}_2(4){:}2$ which is bijective. Moreover, computations in the graph $\Sigma$ give us the following information for two points $y_1, y_2 \in \Sigma _2(x)$ with $Y = \Sigma _1(y_1) \cap \Sigma _1(y_2)$, recorded in Table 1.

Table 1 Intersection patterns in Soicher’s first graph

From Table 1, it follows that for two vertices $y_1, y_2$ in $\Sigma _2(x)$, the involutions $\sigma _{y_1}$ and $\sigma _{y_2}$ are collinear in the near octagon ${\mathbb O}_1(x)$ (which is equivalent to $\sigma _2 \in \mathcal O_{1a}(\sigma _1) \cup \mathcal O_{1b}(\sigma _1)$) if and only if $\mathrm {d}(y_1, y_2) = 4$ or ($\mathrm {d}(y_1, y_2) = 2$ and $|\Sigma _1(x) \cap \Sigma _1(y_1) \cap \Sigma _1(y_2)| = 16$). Thus we have the following.

Theorem 6.1

Let x be a vertex of $\Sigma$. Then the following hold:

• 1. For every vertex $y \in \Sigma _2(x)$, the elementwise stabilizer of $\Sigma _1(x) \cap \Sigma _1(y)$ inside $G_x$ has order 2 and is generated by a central involution $\sigma _y$ of $G_x \cong \mathrm {G}_2(4){:}2$. Moreover, the map $\theta : y \mapsto \sigma _y$ defines a bijection between $\Sigma _2(x)$ and the set of central involutions of $G_x$.

• 2. Let $\Gamma$ denote the graph defined on the set $\Sigma _2(x)$ of vertices at distance 2 from x in $\Sigma$, by making two vertices $y_1, y_2$ adjacent if and only if $\mathrm {d}(y_1, y_2) = 4$ or ($\mathrm {d}(y_1, y_2) = 2$ and $|\Sigma _1(x) \cap \Sigma _1(y_1) \cap \Sigma _1(y_2)| = 16$). Then the map $\theta$ is an isomorphism between $\Gamma$ and the collinearity graph of ${\mathbb O}_1(x)$.

• 3. The map $\theta$ also defines an isomorphism between the subgraph of $\Sigma$ induced on $\Sigma _2(x)$ (the second subconstituent) and the graph obtained from the collinearity graph of ${\mathbb O}_1(x)$ by making two vertices (points of the near octagon) adjacent when they are at distance 2 from each other and have a unique common neighbor.