Difference between revisions of "Locally free group"
From Encyclopedia of Mathematics
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| − | A group in which every finitely-generated subgroup is free (see [[ | + | 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 | + | 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 product of groups|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==== | ====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> | + | <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> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 22:16, 7 November 2016
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
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