J. Korean Math. Soc. 2003; 40(2): 225-239
Printed March 1, 2003
Copyright © The Korean Mathematical Society.
V. V. Bludov, A. M. W. Glass, and Akbar H. Rhemtulla
Irkutsk State University, Centre for Mathematical Sciences, University of Alberta
$(G,<)$ is an ordered group if ‘$<$’ is a total order relation on $G$ in which $f < g$ implies that $xfy < xgy$ for all $f, g, x, y \in G$. We say that $(G,<)$ is centrally ordered if $(G,<)$ is ordered and $[G,D] \subseteq C$ for every convex jump $C \prec D$ in $G$. Equivalently, if $f^{-1}gf \le g^2$ for all $f, g \in G$ with $g > 1$. Every order on a torsion-free locally nilpotent group is central. We prove that if every order on every two-generator subgroup of a locally soluble orderable group $G$ is central, then $G$ is locally nilpotent. We also provide an example of a non-nilpotent two-generator metabelian orderable group in which all orders are central.
Keywords: soluble group, locally nilpotent group, ordered group, convex jump, central series, weakly Abelian
MSC numbers: primary 20F19; secondary 06F15, 20F14, 20E26
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