Difference sets and doubly transitive actions on Hadamard matrices

Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon. n use our classification to show that the only cocyclic Hadamard matrices developed from a difference set with non-affine automorphism group are those that arise from the Paley Hadamard matrices. s a cocyclic Hadamard matrix developed from a difference set then the automorphism group of H is doubly transitive. We classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group. A key component of this is a complete list of difference sets corresponding to the Paley Hadamard matrices. As part of our classification we uncover a new triply infinite family of skew-Hadamard difference sets. To our knowledge, these are the first skew-Hadamard difference sets to be discovered in non-abelian p-groups with no exponent restriction. more application of our main classification, we show that Hallʼs sextic residue difference sets give rise to precisely one cocyclic Hadamard matrix.