RIGHT ENGEL CONDITIONS FOR ORDERABLE GROUPS

Abstract Let g be an element of a group G . For a positive integer n , let $R_n(g)$ be the subgroup generated by all commutators $[\ldots [[g,x],x],\ldots ,x]$ over $x\in G$ , where x is repeated n times. Similarly, $L_n(g)$ is defined as the subgroup generated by all commutators $[\ldots [[x,g],g],\ldots ,g]$ , where $x\in G$ and g is repeated n times. In the literature, there are several results showing that certain properties of groups with small subgroups $R_n(g)$ or $L_n(g)$ are close to those of Engel groups. The present article deals with orderable groups in which, for some $n\geq 1$ , the subgroups $R_n(g)$ are polycyclic. Let $h\geq 0$ , $n&gt;0$ be integers and G be an orderable group in which $R_n(g)$ is polycyclic with Hirsch length at most h for every $g\in G$ . It is proved that there are $(h,n)$ -bounded numbers $h^*$ and $c^*$ such that G has a finitely generated normal nilpotent subgroup N with $h(N)\leq h^*$ and $G/N$ nilpotent of class at most $c^*$ . The analogue of this theorem for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100191_i