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

${G}^{\text{'}}$ 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

${G}^{\text{'}}\cap L=\left\{e\right\}$. 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.