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Graphs and Combinatorics

, Volume 35, Issue 3, pp 579–597 | Cite as

Difference Sets Disjoint from a Subgroup

  • Courtney Hoagland
  • Stephen P. HumphriesEmail author
  • Nathan Nicholson
  • Seth Poulsen
Original Paper
  • 21 Downloads

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.

Keywords

Difference set Subgroup DRAD Schur ring 

Mathematics Subject Classification

Primary 05B10 Secondary 20C05 

Notes

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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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Courtney Hoagland
    • 1
  • Stephen P. Humphries
    • 1
    Email author
  • Nathan Nicholson
    • 1
  • Seth Poulsen
    • 1
  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA

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