Cocyclic Hadamard matrices and Hadamard groups are equivalent

In this paper, we prove that the concepts of cocyclic Hadamard matrix and Hadamard group are equivalent. A general procedure for constructing Hadamard groups and classifying such groups on the basis of isomorphism type is given. To illustrate the ideas, cocyclic Hadamard matrices over dihedral groups are constructed and the corresponding Hadamard groups classified.