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Suppose that $g$ is a primitive root of ${\\mathbb F}_p$. Define the permutation $\\tau_g:\\,{\\mathcal H}_p\\to{\\mathcal H}_p$ by $$ \\tau_g(b):=\\begin{cases} g^b,&\\text{if }g^b\\in{\\mathcal H}_p,\\\\ -g^b,&\\text{if }g^b\\not\\in{\\mathcal H}_p,\\\\ \\end{cases} $$ for each $b\\in{\\mathcal H}_p$, where ${\\mathcal H}_p=\\{1,2,\\ldots,(p-1)/2\\}$ is viewed as a subset of ${\\mathbb F}_p$. In this paper, we investigate the sign of $\\tau_g$. 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