# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] DOI: 10.1007/BF02284589 · Zbl 0275.06018 · doi:10.1007/BF02284589 [2] DOI: 10.4134/JKMS.2003.40.2.225 · Zbl 1031.20036 · doi:10.4134/JKMS.2003.40.2.225 [3] Glass A. M. W., Series in Algebra 7, in: Partially Ordered Groups (1999) [4] DOI: 10.1017/S0305004100037208 · doi:10.1017/S0305004100037208 [5] DOI: 10.1017/S1446788700031001 · doi:10.1017/S1446788700031001 [6] Kokorin A. I., Fully Ordered Groups (1974) · Zbl 0131.26403 [7] DOI: 10.1007/BF02671588 · doi:10.1007/BF02671588 [8] Kopitov V. M., Right-Ordered Groups (1996) [9] DOI: 10.1112/plms/s3-49.1.155 · Zbl 0537.20013 · doi:10.1112/plms/s3-49.1.155 [10] Mura R., Lecture Notes in Pure and Applied Mathematics 27, in: Orderable Groups (1977) [11] Ohnishi M., Osaka J. Math. 2 pp 161– (1950)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.