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Register a new userThe complete splittings of finite abelian groups
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Kevin Zhao
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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 such representation (while $0$ has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our description is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups.
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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 $d_{Omega}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|geq t$ has a subsequence in $Omega$.We provide some first results and open problems on $d_{Omega}(G)$.
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We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called {it nonsingular} if $gcd(|G|, a) = 1$ for any $ain M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits $mathbb{Z}_q$, then there are infinitely many primes $p$ such that $M$ splits $mathbb{Z}_p$.
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