Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2003; 40(2): 225-239

Printed March 1, 2003

Copyright © The Korean Mathematical Society.

Ordered groups in which all convex jumps are central

V. V. Bludov, A. M. W. Glass, and Akbar H. Rhemtulla

Irkutsk State University, Centre for Mathematical Sciences, University of Alberta

Abstract

$(G,<)$ 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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