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On solvable \(R^*\) groups of finite rank. (English) Zbl 1031.20025
Let \(G\) be a group, \(a\in G\) and \(S(a)\) be the semigroup generated by all conjugates of \(a\) in \(G\). It is proved that if \(G\) is a solvable group of finite rank and \(1\notin S(a)\) for all \(1\neq a\in G\), then \(\langle a^G\rangle/\langle b^G\rangle\) is a periodic group for every \(b\in S(a)\). Conversely, if every two generator subgroup of a finitely generated torsion-free solvable group \(G\) has this property then \(G\) has finite rank, and if every finitely generated subgroup has this property then every partial order on \(G\) can be extended to a linear order.

MSC:
20F16 Solvable groups, supersolvable groups
20F60 Ordered groups (group-theoretic aspects)
06F15 Ordered groups
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