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The Neilsen-Schreier Nielsen-Schreier subgroup theorem: subgroups of free groups are free. This has a very quick proof using the fact that a group is free precisely when it acts freely and without inversions on a tree.

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The Neilsen-Schreier subgroup theorem: subgroups of free groups are free. This has a very quick proof using the fact that a group is free precisely when it acts freely and without inversions on a tree.