View source for Locally free group

A group in which every finitely-generated subgroup is free (see [[Finitely-generated group|Finitely-generated group]]; [[Free 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060440/l0604401.png" /> if any finite subset of it is contained in a free subgroup of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060440/l0604402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060440/l0604403.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR></table>