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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
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