Graphs associated with sporadic group geometries, and the semisimple elements of E8(2)

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Abstract

This thesis studies collinearity graphs and commuting involution graphs associated with sporadic group geometries, and the conjugacy classes of semisimple elements in the exceptional Lie-type group E8(2).First we construct plane-line collinearity graphs for the sporadic simple groups and their associated minimal and maximal 2-local parabolic geometries. For such a group and geometry, the plane-line collinearity graph takes all planes of the geometry as its vertices and joins two vertices with an edge if their planes are collinear in the geometry. We construct these graphs for the groups M23, J4, Fi22, Fi23, He, Co3 and Co2. Additionally we construct a variety of collinearity graphs associated with the minimal 2-local geometries of McL.A second short study looks at the commuting involution graphs associated with the Baby Monster sporadic group. These are graphs which take a conjugacy class of involutions as its vertex set and joins two vertices with an edge if they commute. We detail information relating to two such graphs.Finally, we study the conjugacy classes of semisimple elements in the exceptional group E8(2). This study is a joint work with Ali Aubad, John Ballantyne, Alexander McGaw, Peter Neuhaus, Peter Rowley and David Ward in which we determine the structure of centralisers for all such elements including information such as fixed-space dimensions and powering up maps. The ultimate aim is to determine all maximal subgroups of E8(2). This is a lengthy ongoing project and this study forms part of that effort.

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Keyword(s)

E8(2); Lie; collinearity graph; geometry; group; semisimple; sporadic

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