ACTA issues

Difference bases in finite Abelian groups

Taras Banakh, Volodymyr Gavrylkiv

Acta Sci. Math. (Szeged) 85:1-2(2019), 119-137

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 $\eth[G]:=\tfrac{\Delta[G]}{\sqrt{|G|}}$ is called the {\em difference characteristic} of $G$. Using properties of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge11$, any finite Abelian $p$-group $G$ has difference characteristic $\eth[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup _{k\in\IN }\eth[C_{p^k}]< \sqrt{2}\cdot\frac {\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality less than $96$.

DOI: 10.14232/actasm-017-586-x

AMS Subject Classification (1991): 05B10, 05E15, 16L99, 16Z99, 20D60, 20K01

Keyword(s): finite group, Abelian group, difference basis, difference characteristic

Received December 28, 2017 and in final form May 20, 2018. (Registered under 86/2017.)