The coset factorization of finite cyclic group

Abstract

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 $g=a+b,a\in A,b\in B$. 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 $A=[0,n-k-1]\cup \{i_0,i_1,\dots i_{k-1}\}$ with $|A|=n$ and $n\ge 2k+1$. We obtain the following conclusion: If $(i)$ $k\le 2$ 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 $modulo$ $n$.