Su di un problema combinatorio in teoria dei gruppi

Curzio, Mario ; Longobardi, Patrizia ; Maj, Mercede

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 74 (1983) no. 3, pp. 136-142

Résumé

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 $x_{1},x_{2},\cdots,x_{n}$ in $G$, there exists some permutation $\sigma$ of $\{ 1,2,\cdots,n \}$, $\sigma \ne id.$ such that $x_{1},x_{2},\cdots,x_{n} = x_{\sigma(1)},x_{\sigma(2)},\cdots,x_{\sigma(n)}$. 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.