×

zbMATH — the first resource for mathematics

Normal relatively convex subgroups of solvable orderable groups. (English. Russian original) Zbl 1245.20049
Algebra Logic 48, No. 3, 163-172 (2009); translation from Algebra Logika 48, No. 3, 291-308 (2009).
Summary: Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed, then it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group \(G\) of length three with periodic \(G/G'\) and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed.

MSC:
20F60 Ordered groups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20F16 Solvable groups, supersolvable groups
06F15 Ordered groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. M. Kopytov and N. Ya. Medvedev, Right Ordered Groups [in Russian], Nauch. Kniga, Novosibirsk (1996). · Zbl 0896.06017
[2] Unsolved Problems in Group Theory, The Kourovka Notebook, 16th edn., Institute of Mathematics SO RAN, Novosibirsk (2006); http://www.math.nsc.ru/\(\sim\)alglog . · Zbl 1155.20309
[3] A. Rhemtulla and H. Smith, ”On solvable R* groups of finite rank,” Com. Alg., 31, No. 7, 3287–3293 (2003). · Zbl 1031.20025 · doi:10.1081/AGB-120022225
[4] A. I. Kokorin and V. M. Kopytov, Linearly Ordered Groups [in Russian], Nauka, Moscow (1972). · Zbl 0192.36401
[5] R. B. Mura and A. H. Rhemtulla, Orderable Groups, Lect. Notes Pure Appl. Math., 27, Marcel Dekker, New York (1977). · Zbl 0358.06038
[6] P. H. Kropholler, ”On finitely generated soluble groups with no large wreath product sections,” Proc. London Math. Soc., III. Ser., 49, No. 1, 155–169 (1984). · Zbl 0537.20013 · doi:10.1112/plms/s3-49.1.155
[7] Yu. M. Gorchakov, ”An example of G-periodical torsion-free group,” Algebra Logika, 6, No. 3, 5–7 (1967). · Zbl 0167.28704
[8] D. M. Smirnov, ”Right ordered groups,” Algebra Logika, 5, No. 6, 41–59 (1966).
[9] A. I. Kokorin, ”\(\Gamma\)-O*-subgroups and relatively convex subgroups of ordered groups,” Sib. Mat. Zh., 7, No. 3, 713–717 (1966).
[10] N. Ya. Medvedev, ”Soluble groups and varieties of -groups,” Algebra Logika, 44, No. 3, 355–367 (2005). · Zbl 1101.06011
[11] A. M. Glass and N. Ya. Medvedev, ”Unilateral o-groups,” Algebra Logika, 45, No. 1, 20–27 (2006). · Zbl 1115.06008 · doi:10.1007/s10469-006-0002-y
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.