Abstract
We study the notion of formal duality as introduced by Cohn, Kumar, Reiher and Schürmann. It can be reduced to a study of formally dual subsets in finite Abelian groups. We prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order \(2^{2l+1}\), there is no primitive pair of formally dual subsets. This partially proves a conjecture that the only cyclic groups with a pair of primitive formally dual subsets are \(\{0\}\) and \({\mathbb {Z}}/4{\mathbb {Z}}\). On the way, we obtain several necessary properties about formally dual subsets of general finite cyclic groups.
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Acknowledgements
I like to thank Erik Friese for several inspirations, pointers to literature and verifications of arguments. Moreover I am grateful to Frieder Ladisch for plenty of helpful remarks and inspiring a shorter, more elegant proof. Furthermore I like to express my sincere gratitude to Achill Schürmann for suggesting this interesting question as well as many patient reviews and for pointing out several mistakes in preliminary versions of this paper.
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Schüler, R. Formally dual subsets of cyclic groups of prime power order. Beitr Algebra Geom 58, 535–548 (2017). https://doi.org/10.1007/s13366-017-0337-7
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DOI: https://doi.org/10.1007/s13366-017-0337-7