**Any subgroup of a free group is free. **

This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X.

The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore its fundamental group H is free.

*Any free group may be realized as the fundamental group of a graph X.* Let G be the free group on *n* generators. By the Seifert-van Kampen theorem, G is the fundamental group of the graph G’ consisting of a single vertex *v* and* n* loops incident with *v *(i.e. the bouquet of *n* circles.) Note that any connected graph may be realized as a bouquet, by taking the quotient by a spanning tree.

*The main theorem on covering spaces*

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