Abstract
We study finite groups G having a subgroup H and \(D \subset G {\setminus } H\) such that (i) the multiset \(\{ xy^{-1}:x,y \in D\}\) has every element that is not in H occur the same number of times (such a D is called a relative difference set); (ii) \(G=D\cup D^{(-1)} \cup H\); (iii) \(D \cap D^{(-1)} =\emptyset \). We show that \(|H|=2\), that H is central and that G is a group with a single involution. We also show that G cannot be abelian. We give infinitely many examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito.
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Acknowledgements
All computations made in the preparation of this paper were accomplished using Magma [23]. The first, second, third, and fifth authors thank Brigham Young University Department of Mathematics for funding during the writing of this paper. We are also grateful for useful suggestions from a referee.
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The first, second, third, and fifth authors thank Brigham Young University Department of Mathematics for funding during the writing of this paper.
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Anderson, G., Haviland, A., Holmes, M. et al. Difference Sets Disjoint from a Subgroup III: The Skew Relative Cases. Graphs and Combinatorics 39, 67 (2023). https://doi.org/10.1007/s00373-023-02662-8
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DOI: https://doi.org/10.1007/s00373-023-02662-8