Abelian and non-abelian Paley type group schemes

Abstract

In this paper, we present a construction of abelian Paley type group schemes which are inequivalent to Paley group schemes. We then determine the equivalence amongst their configurations, the Hadamard designs or the Paley type strongly regular graphs obtained from these group schemes, up to isomorphism. We also give constructions of several families of non-abelian Paley type group schemes using strong multiplier groups of the abelian Paley type group schemes, and present the first family of p-groups of non-square order and of non-prime exponent that contain Paley type group schemes for all odd primes p.