Difference bases in dihedral groups

Document Type: Research Paper

Authors

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

Abstract

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 $\le80$‎.

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