Geodesic
Groups Generated by (near) Mutually Engel Periodic Pairs
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 485-497.

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We use notations: $[x, y]=[x_{,1} y]$ and $[x_{,k+1} y]$ e $[[x_{,k} y], y]$. We consider groups generated by $x$, $y$ satisfying relations $x = [x_{,n} y], y = [y_{,n} x]$ or $[x, y]=[x_{,n} y]$, $[y, x]=[y_{,n} x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $Sl_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $\langle x,y \mid [x,y] = [x_{,2} y], [y,x]=[y_{,2} x], x^n, y^m \rangle$ are finite.
Scriviamo $[x, y]=[x_{,1} y]$ e $[x_{,k+1} y]$ e $[[x_{,k} y], y]$. Nel presente mostriamo certe proprietà ed esempio dei gruppi con i generatori $x$, $y$ tali che $x = [x_{,n} y], y = [y_{,n} x]$ o $[x, y]=[x_{,n} y]$, $[y, x]=[y_{,n} x]$. 
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     author = {S{\l}anina, Piotr and Tomaszewski, Witold},
     title = {Groups {Generated} by (near) {Mutually} {Engel} {Periodic} {Pairs}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {485--497},
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     volume = {Ser. 8, 10B},
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     year = {2007},
     zbl = {1167.20018},
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     language = {en},
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Słanina, Piotr; Tomaszewski, Witold. Groups Generated by (near) Mutually Engel Periodic Pairs. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 485-497. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_2_a15/

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