Graphs related to E7(q)

Chris Krook

Distance-transitive graphs stand out from the crowd by their high degree of symmetry. Thanks to this symmetry such graphs can be described very efficiently. A distance-transitive graph is determined uniquely by its automorphism group, a corresponding maximal subgroup that stabilizes a vertex in the graph, and an arbitrary edge. The question arises for which simple groups and corresponding maximal subgroups, there exists a distance-transitive graph in which these groups play the roles as described above. Using the classification of all finite simple groups, many researchers have tried to answer this question. Much work has already been done. There exists, however, a small list with hard cases, that still need to be investigated. The first part of my talk will be aimed at giving a nice introduction into graphs, and distance-transitive graphs in particular, to laymen. The goal is to explain what I have been working on to non-mathematicians. Concepts like groups will be introduced, while keeping the mathematical notations to a minimum. In the second part of this talk I will examine one of the hard cases mentioned above. I create a general graph structure with automorphism group E7(q) and vertex-stabilizer A7(q)· 2 over a field of characteristic 2. Using character theory, I show that such a graph cannot be distance-transitive. In this manner, I contributed to the classification of all primitive distance-transitive graphs.

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