an:01952967
Zbl 1031.20025
Rhemtulla, Akbar; Smith, Howard
On solvable \(R^*\) groups of finite rank
EN
Commun. Algebra 31, No. 7, 3287-3293 (2003).
00098538
2003
j
20F16 20F60 06F15
solvable groups of finite rank; orderable groups; partial orders; linear orders; finitely generated subgroups
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.
Nikolai Yakovlevich Medvedev (Barnaul)