On Relative Difference Sets in Dihedral Groups

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 af...

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Main Authors: Garciano, Agnes, Hiramine, Yutaka, Yokonuma, Takeo
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Published: Archīum Ateneo 2006
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Online Access:https://archium.ateneo.edu/mathematics-faculty-pubs/90
https://link.springer.com/article/10.1007%2Fs10623-005-2399-z
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Institution: Ateneo De Manila University
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spelling ph-ateneo-arc.mathematics-faculty-pubs-10892020-06-18T07:43:42Z On Relative Difference Sets in Dihedral Groups Garciano, Agnes Hiramine, Yutaka Yokonuma, Takeo 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. 2006-01-01T08:00:00Z text https://archium.ateneo.edu/mathematics-faculty-pubs/90 https://link.springer.com/article/10.1007%2Fs10623-005-2399-z Mathematics Faculty Publications Archīum Ateneo relative difference sets dihedral groups affine type Mathematics
institution Ateneo De Manila University
building Ateneo De Manila University Library
country Philippines
collection archium.Ateneo Institutional Repository
topic relative difference sets
dihedral groups
affine type
Mathematics
spellingShingle relative difference sets
dihedral groups
affine type
Mathematics
Garciano, Agnes
Hiramine, Yutaka
Yokonuma, Takeo
On Relative Difference Sets in Dihedral Groups
description 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.
format text
author Garciano, Agnes
Hiramine, Yutaka
Yokonuma, Takeo
author_facet Garciano, Agnes
Hiramine, Yutaka
Yokonuma, Takeo
author_sort Garciano, Agnes
title On Relative Difference Sets in Dihedral Groups
title_short On Relative Difference Sets in Dihedral Groups
title_full On Relative Difference Sets in Dihedral Groups
title_fullStr On Relative Difference Sets in Dihedral Groups
title_full_unstemmed On Relative Difference Sets in Dihedral Groups
title_sort on relative difference sets in dihedral groups
publisher Archīum Ateneo
publishDate 2006
url https://archium.ateneo.edu/mathematics-faculty-pubs/90
https://link.springer.com/article/10.1007%2Fs10623-005-2399-z
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