On Relative Difference Sets in Dihedral Groups |
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Authors: | Agnes D Garciano Yutaka Hiramine Takeo Yokonuma |
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Institution: | (1) Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, The Philippines;(2) Department of Mathematics, Faculty of Education, Kumamoto University, Kurokami, Kumamoto, Japan;(3) Department of Mathematics, Sophia University, Kioicho, Chiyoda-ku, Tokyo, Japan |
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Abstract: | In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters
of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference
set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral
groups and Menon–Hadamard difference sets.
In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if
a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes. |
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Keywords: | relative difference sets dihedral groups affine type |
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