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Free groups and topology

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 (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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