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Nonexistence of abelian difference sets: Lander's conjecture for prime power orders


Authors: Ka Hin Leung, Siu Lun Ma and Bernhard Schmidt
Journal: Trans. Amer. Math. Soc. 356 (2004), 4343-4358
MSC (2000): Primary 05B10; Secondary 05B20
DOI: https://doi.org/10.1090/S0002-9947-03-03365-8
Published electronically: August 26, 2003
MathSciNet review: 2067122
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Abstract: In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order $>4$ and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow $p$-subgroup containing a difference set of order divisible by $p$. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.


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Additional Information

Ka Hin Leung
Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore
Email: matlkh@nus.edu.sg

Siu Lun Ma
Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore
Email: matmasl@nus.edu.sg

Bernhard Schmidt
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
Email: schmidt@math.uni-augsburg.de

DOI: https://doi.org/10.1090/S0002-9947-03-03365-8
Keywords: Difference set, Ryser's conjecture, Lander's conjecture, field descent
Received by editor(s): November 13, 2002
Received by editor(s) in revised form: April 10, 2003
Published electronically: August 26, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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