A Recursive Construction for Skew Hadamard Difference Sets

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.