^{2}Ivan Franko National University of Lviv (Ukraine), and Institute of Mathematics, Jan Kochanowski University in Kielce (Poland)

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 $n\in\IN$ 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 $\le80$.