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A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ 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 $pge 11$, any finite Abelian $p$-group $G$ has difference characteristic $eth[G]<frac{sqrt{p}-1}{sqrt{p}-3}cdotsup_{kinmathbb N}eth[C_{p^k}]<sqrt{2}cdotfrac{sqrt{p}-1}{sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.
A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=a-b$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the diffe
A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the difference
We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without
Let $G$ be a finite group. We will say that $M$ and $S$ form a textsl{complete splitting} (textsl{splitting}) of $G$ if every element (nonzero element) $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, and $0$ has a
The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $Omega subset B(G$), let $d_{O