# Locally cyclic group

From Encyclopedia of Mathematics

2010 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,
*The Theory of Groups*, reprinted American Mathematical Society (1976) ISBN 0-8218-1967-4

**How to Cite This Entry:**

Locally cyclic group.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Locally_cyclic_group&oldid=35797