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Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.

Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.