论文标题
有限循环组的固定分解
The coset factorization of finite cyclic group
论文作者
论文摘要
让$ g $成为一个有限的循环群，并添加书面，让$ a，\ b $为$ g $的非发空子集。我们会说$ g = a+b $是a \ textit {cresisization}，如果对于$ g $中的每个$ g $ in $ g $，则有unique元素$ a，\ b $ of $ g $，以便$ g = a+b，\ b，\ a \ in a，b \ in b $中。特别是，如果$ a $是一组完整的残基集$ modulo $ $ | a | $，那么我们将分解称为$ g $的textit {coset cressization}。在本文中，我们主要研究一个分解$ g = a+b $，其中$ g $是有限的环状组，$ a = [0，n-k-1] \ cup \ {i_0，i_1，i_1，\ ldots i_ {k-1} \} $，带有$ | = n $ | = n $和$ n $和$ n $ and $ n $ and \ geq 2k 2k+1 $ 1 $ 1 $。我们得出以下结论：如果$（i）$ $ k \ leq 2 $或$（ii）$ $ gcd（| a |，| b |）$的独特prime除法数量最多是$ 1 $或$（iii）$ $ gcd（| a | a |，| b |）= pq $ with with pq $ $ gcd（pq，\ frac {| b |} {gcd（| a |，| b |）}）= 1 $，然后$ a $是完整的残基集$ modulo $ $ $ n $。
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 $A=[0,n-k-1]\cup\{i_0,i_1,\ldots i_{k-1}\}$ with $|A|=n$ and $n\geq 2k+1$. We obtain the following conclusion: If $(i)$ $k\leq 2$ 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 $modulo$ $n$.
