Using the 11-point biplane and L2(11) to understand J1

Abstract

In this paper, we use the 11-point biplane and its automorphisms in $L_2(11)$ to label and study the Livingstone graph ($\mathrm{\Gamma }$) and $J_1$, with an aim of using the simplest methods possible. We detail the action of $J_1$ on $\mathrm{\Gamma }$, along with the adjacencies and coadjacencies (vertices at maximum distance) in $\mathrm{\Gamma }$. In the last section, we use this apparatus to describe the generation of subgroups of the form $2^3:7:3$ and an elegant substructure of $\mathrm{\Gamma }$ fixed by a maximal subgroup of $J_1$ isomorphic to $19:6$.