USING THE 11-POINT BIPLANE AND L-2(11) TO UNDERSTAND J(1)

摘要

In this paper, we use the 11-point biplane and its automorphisms in L-2 (11) to label and study the Livingstone graph (Gamma) and J(1), with an aim of using the simplest methods possible. We detail the action of J(1) on Gamma, along with the adjacencies and coadjacencies (vertices at maximum distance) in 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 Gamma fixed by a maximal subgroup of J(1) isomorphic to 19 : 6.