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In this paper we characterize certain classes of groups $G$ in which, from $x^p=y^p$ ($x,y\in G$, $p$ a fixed prime), it follows that $xy=yx$. Our results extend results previously obtained by other authors, in the finite case.

M. V. Sapir ha formulato la seguente congettura: non esiste un semigruppo $S$ infinito, finitamente generabile, soddisfacente l'identità $x^2=0$ e immagine omomorfa di un sottosemigruppo di un gruppo $G$ nilpotente. Se ciò vale, ogni gruppo risolubile con una base finita per le sue identità semigruppali è abeliano o di esponente finito. In questo lavoro si prova la congettura di Sapir quando l'interderivato ${\gamma }_3\left(G\right)$ è periodico o se $S$ è $3$-generato e ${\gamma }_4\left(G\right)$ è periodico.

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.

A group $G$ in a variety $\mathfrak{V}$ is said to be absolutely-$\mathfrak{V}$, and we write $G\in A\mathfrak{V}$, if central extensions by $G$ are again in $\mathfrak{V}$. Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class $A{\mathfrak{N}}_k$ of absolutely-nilpotent of class $k$ groups. We prove some closure properties of the class $A{\mathfrak{N}}_k$ and we show that every nilpotent of class $k$ group can be embedded in an $A{\mathfrak{N}}_k$-gvoup. We describe all metacyclic $A{\mathfrak{N}}_k$-groups and we characterize $2$-generator and infinite $3$-generator $A{\mathfrak{N}}_2$-groups. Finally...