# Locally free group

From Encyclopedia of Mathematics

A group in which every finitely-generated subgroup is free (see Finitely-generated group; Free group). Thus, a countable locally free group is the union of an ascending sequence of free subgroups.

One says that a locally free group has finite rank $n$ if any finite subset of it is contained in a free subgroup of rank $n$, $n$ being the smallest number with this property. The class of locally free groups is closed with respect to taking free products, and the rank of a free product of locally free groups of finite ranks equals the sum of the ranks of the factors.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

**How to Cite This Entry:**

Locally free group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_free_group&oldid=39697

This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article