Difference bases in dihedral groups

Document Type : Research Paper


1 Ivan Franko National University of Lviv (Ukraine), and Institute of Mathematics, Jan Kochanowski University in Kielce (Poland)

2 Vasyl Stefanyk Precarpathian National‎ ‎University‎, ‎Ivano-Frankivsk‎, ‎Ukraine


A subset $B$ of a group $G$ is called a {\em‎ ‎difference basis} of $G$ if each element $g\in G$ can be written as the‎ ‎difference $g=ab^{-1}$ of some elements $a,b\in B$‎. ‎The smallest‎ ‎cardinality $|B|$ of a difference basis $B\subset G$ is called the {\em‎ ‎difference size} of $G$ and is denoted by $\Delta[G]$‎. ‎The fraction ‎‎‎$\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the {\em difference characteristic} of $G$‎. ‎We prove that for every $n\in N$ the dihedral group‎ ‎$D_{2n}$ of order $2n$ has the difference characteristic‎ ‎$\sqrt{2}\le\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$‎. ‎Moreover‎, ‎if $n\ge 2\cdot 10^{15}$‎, ‎then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$‎. ‎Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$‎.


Main Subjects

[1] T. Banakh and V. Gavrylkiv, Algebra in the sup erextensions of twinic groups, Dissertationes Math. 473 (2010) pp.
[2] T. Banakh and V. Gavrylkiv, Difference bases in cyclic groups, em J. Algebra Appl. bf 18 (2019) pp. 18.
[3] T. Banakh, V. Gavrylkiv and O. Nykyforchyn, Algebra in sup erextension of groups, I: zeros and commutativity,
Algebra Discrete Math. , 3 (2008) 1{29.
[4] R. C. Bose, An affine analogue of Singer's theorem, J. Indian Math. Soc. , 6 (1942) 1{15.
[5] R. C. Bose and S. Chowla, Theorems in the additive theory of numb ers, Comment. Math. Helv. , 37 (1962-63)
[6] P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers , Ph.D. Thesis, Univ. de Limoges, 1998.
[7] V. Gavrylkiv, Bases in dihedral and Bo olean groups, J. Integer Seq. , 20 (2017).
[8] M. Golay, Notes on the representation of 1 ; 2 ; : : : ; n by differences, J. Lond. Math. Soc. , 4 (1972) 729{734.
[9] J. Leech, On the representation of 1 ; 2 ; : : : ; n by differences, J. Lond. Math. Soc. , 31 (1956) 160{169.
[10] G. Kozma and A. Lev, Bases and decomp osition numb ers of nite groups, Arch. Math. (Basel) , 58 (1992) 417{424.
[11] L. Redei and A. Renyi, On the representation of the numb ers 1 ; 2 ; : : : ; N by means of differences, Math. Sbornik
N.S. , 24 (1949) 385{389.
[12] I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith. , 3 (1993) 259{282.
[13] J. Singer, A theorem in nite pro jective geometry and some applications to numb er theory, Trans. Amer. Math.
Soc. , 43 (1938) 377{385.