Classification of Flag-Transitive Steiner Quadruple Systems

A Steiner quadruple system of order v is a 3−(v, 4, 1) design and will be denoted SQS(v). Using the classification of finite 2-transitive permutation groups all SQS(v) with a flag-transitive automorphism group are completely classified, thus solving the “still open and longstanding problem of classifying all flag-transitive 3−(v, k, 1) designs” for the smallest value of k. Moreover, a generalization of a result of H. Lüneburg (1965, Math. Z.89, 82–90) is achieved.