Difference bases in finite Abelian groups

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 \(\eth[G]:=\frac{\Delta[G]}{\sqrt{|G|}}\) is called the difference characteristic of \(G\). Using properies 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 \(p\ge 11\), any finite Abelian \(p\)-group \(G\) has difference characteristic \(\eth[G]