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Difference Sets Disjoint from a Subgroup

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Abstract

We study finite groups G having a non-trivial, proper subgroup H and \(D \subset G {\setminus } H, D \cap D^{-1}=\emptyset ,\) such that the multiset \(\{ xy^{-1}:x,y \in D\}\) has every non-identity element occur the same number of times (such a D is called a difference set). We show that \(|G|=|H|^2\), and that \(|D \cap Hg|=|H|/2\) for all \(g \notin H\). We show that H is contained in every normal subgroup of index 2, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets, and that, further, these difference sets determine DRADs.

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Acknowledgements

We are grateful to Pace Nielsen for useful conversations regarding this paper, and also to an anonymous referee for useful comments, of notifying us of reference [5] and a proof that \(m=0\). We thank another referee for a simplification of the proof of Theorem 1.1. All calculations made in the preparation of this paper were accomplished using Magma [1].

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Correspondence to Stephen P. Humphries.

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Hoagland, C., Humphries, S.P., Nicholson, N. et al. Difference Sets Disjoint from a Subgroup . Graphs and Combinatorics 35, 579–597 (2019). https://doi.org/10.1007/s00373-019-02017-2

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