Properties of locally cyclic groups

Locally cyclic group is a group every finite set of elements of which generates a cyclic subgroup. We give examples of periodic locally cyclic groups and locally cyclic torsion-free groups. Properties of locally cyclic groups are studied. A locally cyclic group cannot be mixed, that is, it cannot contain elements of finite and infinite order simultaneously. A locally cyclic group is Abelian. By their properties periodic locally cyclic groups and locally cyclic torsion-free groups are distinguished. The Sylow subgroups of a periodic locally cyclic group are cyclic or quasi-cyclic. A periodic locally cyclic group decomposes into a direct product of Sylow subgroups. By N. F. Sesekin and A. I. Starostin the fol- lowing theorem is proved: a locally finite group, all Sylow p-subgroups of which are quasi-cyclic, is a complete peri- odic locally cyclic group. Here, in addition to this theorem, we consider the structure of a complete periodic locally cyclic group. A complete periodic locally cyclic group decomposes into a direct product of quasi-cyclic subgroups with distinct prime numbers. A complete periodic locally cyclic group is uniquely reconstructed by its lower layer. In this article an example is given of the fact that an arbitrary periodic locally cyclic group is not unique reconstructed by its lower layer. A torsion-free locally cyclic group is isomorphic to a subgroup of the additive group of rational numbers. A periodic locally cyclic group is layer-finite, that is a number of it’s elements of each order is finite. A locally cyclic group can be either a layer-finite or a subgroup of additive groups of rational numbers. The results can be applied when encoding information in space communications.