Every subgroup of a free group is itself a free group. The theorem follows by representing the subgroup by a connected covering graph of a rose and choosing a maximal subtree.
If has finite index in the free group , thenIndeed, the -sheeted covering graph of the rose with petals has vertices and edges, so its fundamental group has rank .
For finite-index subgroups of a rank- free group, the Nielsen–Schreier formula gives the displayed quotient. Along any sequence whose finite indices tend to infinity, the quotient tends to . Subtracting one from the rank before dividing gives the constant rank gradient .
// Target: geometry-and-topology.bigb
If has finite index in a finitely generated group , then Schreier's lemma givesEquality holds when is a free group of finite rank.
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The Nielsen–Schreier theorem is a result in group theory that provides a characterization of free groups in terms of their subgroups. The theorem states that every subgroup of a free group is free. More specifically, if \( F \) is a free group, then any subgroup \( H \) of \( F \) is itself a free group, possibly on a different set of generators.