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We show that there is a permutation $\\sigma\\in S_{n-1}$ such that all the elements $sa_{\\sigma(s)}\\ (s=1,\\ldots,n-1)$ are nonzero if and only if $$\\left|\\left\\{1\\le s<n:\\ \\frac{n}{d}a_s\\ne 0\\right\\}\\right|\\ge d-1\\ \\ \\textrm{ for every positive divisor }\\ d\\ \\textrm{ of }\\ n.$$ When $G$ is the cyclic group $\\mathbb Z/n\\mathbb Z$, this confirms a conjecture of Z.-W. 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We show that there is a permutation $\\sigma\\in S_{n-1}$ such that all the elements $sa_{\\sigma(s)}\\ (s=1,\\ldots,n-1)$ are nonzero if and only if $$\\left|\\left\\{1\\le s<n:\\ \\frac{n}{d}a_s\\ne 0\\right\\}\\right|\\ge d-1\\ \\ \\textrm{ for every positive divisor }\\ d\\ \\textrm{ of }\\ n.$$ When $G$ is the cyclic group $\\mathbb Z/n\\mathbb Z$, this confirms a conjecture of Z.-W. 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