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Register a new userThe coset factorization of finite cyclic group
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Kevin Zhao
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Let $G$ be a finite cyclic group, written additively, and let $A, B$ be nonempty subsets of $G$. We will say that $G= A+B$ is a textit{factorization} if for each $g$ in $G$ there are unique elements $a, b$ of $G$ such that $g=a+b, ain A, bin B$. In particular, if $A$ is a complete set of residues $modulo$ $|A|$, then we call the factorization a textit{coset factorization} of $G$. In this paper, we mainly study a factorization $G= A+B$, where $G$ is a finite cyclic group and $A=[0,n-k-1]cup{i_0,i_1,ldots i_{k-1}}$ with $|A|=n$ and $ngeq 2k+1$. We obtain the following conclusion: If $(i)$ $kleq 2$ or $(ii)$ The number of distinct prime divisors of $gcd(|A|,|B|)$ is at most $1$ or $(iii)$ $gcd(|A|,|B|)=pq$ with $gcd(pq,frac{|B|}{gcd(|A|,|B|)})=1$, then $A$ is a complete set of residues $modulo$ $n$.
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