A subset $B$ of a group $G$ is called a 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 difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{ \sqrt{ |G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ 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 $\le80$.
Categories: math.CO math.GR