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Ordered groups in which all convex jumps are central. (English) Zbl 1031.20036
A linearly ordered group \(G\) is centrally ordered if \([G,D]\subseteq C\) for every convex jump \(C\prec D\) in \(G\) or equivalently, if \(f^{-1}gf\leq f^2\) for all \(f,g\) with \(g>1\). It is proved that if every order on every two-generator subgroup of a locally soluble orderable group is central, then \(G\) is locally nilpotent. An example of a non-nilpotent two-generator metabelian orderable group in which all orders are central is constructed.

MSC:
20F60 Ordered groups (group-theoretic aspects)
06F15 Ordered groups
20F19 Generalizations of solvable and nilpotent groups
20F14 Derived series, central series, and generalizations for groups
20E26 Residual properties and generalizations; residually finite groups
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