John led the discussion again this week and looked at coset geometries and polar spaces.

John began with the following pictureThis is an incidence geometry with 15 points and 15 lines each containing 3 points. It is the smallest thick generalised quadrangle and denoted by $W(3,2)$. It has automorphism group $S_6$. This can be seen by taking the points to be the edges of the complete graph $K_6$ and the lines to be the matchings of $K_6$ (that is the sets of three disjoint edges). The incidence graph of this geometry is known as Tutte’s 8-cage or the Tutte-Coxeter graph.

John next drew the Fano planeThis projective plane is denoted by $PG(3,2)$ and is the smallest thick generalised triangle. It has automorphism group $PGL(3,2)\cong PSL(2,7)$. The incidence graph of this geometry is the Heawood graph.

Next we had the smallest (thick) generalised digon, which is the geometry with three points and three lines such that each line consists of all three points. The incidence graph for this geometry is the complete bipartite graph $K_{3,3}$.