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On a criterion for cyclic orderability of a group. (Russian) Zbl 0713.20034
The authors give the following criterion for cyclic orderability of a group G: The torsion subgroup T(G) of G can be embedded in the toroidal group (i.e. the multiplicative group of complex numbers of module 1), b) G/T(G) is linearly orderable, and c) The commutator subgroup G ' of G does not contain periodic elements. Corollaries. An abelian group is cyclically orderable iff its torsion subgroup is cyclically orderable. Two sufficient conditions for the cyclic orderability of a group G: 1. For each cyclically ordered normal subgroup L of G the factor group G/L is linearly ordered and G ' L={e}. 2. For each G-linearly ordered normal subgroup L of G the factor group G/L is linearly ordered. The authors give an example of a cyclically unorderable group, which shows that the criterion for cyclic orderability given in a paper by S. D. Zheleva [Sib. Mat. Zh. 17, No.5, 1046-1051 (1976; Zbl 0362.06022)] is false.
Reviewer: F.Šik
MSC:
20F60Ordered groups (group aspects)
06F15Ordered groups
20E07Subgroup theorems; subgroup growth