Locally cyclic group

2020 Mathematics Subject Classification: Primary: 20E [MSN][ZBL]

A group in which every finitely generated subgroup is cyclic. In such a group, either every element is of finite order (periodic), or no element other than the identity is (aperiodic). The additive group of rational numbers $\mathbb{Q}^+$ is an aperiodic example, and the group $\mathbb{Q}/\mathbb{Z}$ is a periodic example. The lattice of subgroups of a group is a distributive lattice if and only if the group is locally cyclic.

References

• Marshall Hall jr, The Theory of Groups, reprinted American Mathematical Society (1976)[1959] ISBN 0-8218-1967-4 Zbl 0084.02202 Zbl 0354.20001