Groups Generated by (near) Mutually Engel Periodic Pairs

Słanina, Piotr ; Tomaszewski, Witold

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 2, pp. 485-497

Résumé

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.