Factorisation of Finite Groups and Graphs

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The combinatorial structure of a graph is strongly influenced by its symmetry, and the symmetry is described precisely by its group of automorphisms. Interplay between actions of the automorphism group on vertices, edges, and other configurations, reveals important graph structure, especially the existence of graph factorisations. In turn, a group factorisation arises whenever a group has two independent transitive actions, and these arise in particular while determining graph automorphism groups, and graph factorisations. We will classify families of group factorisations, especially for simple groups, and apply this to establish a theory of symmetrical graph factorisations, and to study Cayley graphs and 2-closures of permutation groups.