Abstract
The fact that the groups $${\mathbb {Z}}_{2^m} \times {\mathbb {Z}}_{2^m}$$ contain difference sets was first established by induction by Jim Davis in his Virginia dissertation of 1987. Later that year we gave a direct construction for a very large family of highly structured inequivalent difference sets in these groups. In this paper, we give a proof of our result which we presented long ago in a colloquium at Wright State University, but which has never been published. While the proof lays bare the rich structure of the difference sets, its utter simplicity suggests that it might just be from the book. We also provide some historical background and elaborate on some of the structural properties of these difference sets which have motivated some recent progress toward the goal of classifying all hadamard 2-groups.
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