Inequivalence of difference sets: on a remark of Baumert (Q1953424)

Summary: An often cited statement of \textit{L. D. Baumert} in his book [Cyclic difference sets (1971; Zbl 0218.05009)] asserts that four well known families of cyclic (\(4t-1,2t-1,t-1\))-difference sets are inequivalent, apart from a small number of exceptions with \(t< 8\). We are not aware of a proof of this statement in the literature. Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases.