Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ORDERED GROUPS IN WHICH ALL CONVEX JUMPS ARE CENTRAL

  • Bludov, V.V. (Institute of Mathematics and Economics Irkutsk State University) ;
  • Glass, A.M.W. (Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences) ;
  • Rhemtulla, Akbar H. (Department of Mathematical and Statistical University of Alberta)
  • Published : 2003.03.01
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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 $f^{-1}g f{\leq} g^2$ 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.

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  2. On SolvableR* Groups of Finite Rank vol.31, pp.7, 2003, https://doi.org/10.1081/AGB-120022225