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We also show that $n\\mid\\text{per}[i^{j-1}]_{1\\le i,j\\le n}$ for any integer $n>2$. We conjecture that if a group $G$ contains no element of order among $2,\\ldots,n+1$ then any $A\\subseteq G$ with $|A|=n$ can be written as $\\{a_1,\\ldots,a_n\\}$ with $a_1,a_2^2,\\ldots,a_n^n$ pairwise distinct. 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In particular, we prove that there is a unique permutation $\\pi$ of $\\{1,\\ldots,n\\}$ such that all the numbers $k+\\pi(k)$ ($k=1,\\ldots,n$) are powers of two. We also show that $n\\mid\\text{per}[i^{j-1}]_{1\\le i,j\\le n}$ for any integer $n>2$. We conjecture that if a group $G$ contains no element of order among $2,\\ldots,n+1$ then any $A\\subseteq G$ with $|A|=n$ can be written as $\\{a_1,\\ldots,a_n\\}$ with $a_1,a_2^2,\\ldots,a_n^n$ pairwise distinct. 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