@article {
author = {Banakh, Taras and Gavrylkiv, Volodymyr},
title = {Difference bases in dihedral groups},
journal = {International Journal of Group Theory},
volume = {8},
number = {1},
pages = {43-50},
year = {2019},
publisher = {University of Isfahan},
issn = {2251-7650},
eissn = {2251-7669},
doi = {10.22108/ijgt.2017.21612},
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 $\le80$.},
keywords = {dihedral group,difference basis,difference characteristic},
url = {https://ijgt.ui.ac.ir/article_21612.html},
eprint = {https://ijgt.ui.ac.ir/article_21612_09abda43cc316d9fccf8681b5cc2872d.pdf}
}