RELATIONS ADMITTING A TRANSITIVE GROUP OF AUTOMORPHISMS

Abstract

The concepts of a Cayley relation of arbitrary arity and a quotient relation are defined. Cayley relations are characterized as those relations whose automorphism groups contain regular subgroups. The freedom of Cayley relations is proved: any relation with a transitive automorphism group is isomorphic to a quotient relation of a Cayley relation.Using Cayley relations, two problems are solved: 1) for a given transitive permutation group on a set to construct all relations on whose automorphism groups contain it; 2) for a given abstract group to construct all relations whose automorphism groups contain a transitive subgroup isomorphic to .Cayley relations are used to describe the graphs, digraphs, and tournaments having a transitive automorphism group. A solution is given for a weak variant of a problem of Konig: what is the nature of a transitive permutation group if there exists a nontrivial graph whose automorphism group contains ?Finally, Cayley relations are used to describe the centralizer of a transitive permutation group in the symmetric group.Bibliography: 23 titles.