Nielsen–Schreier theorem

In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.

A free group may be defined from a group presentation consisting of a set of generators and the empty set of relations (equations that the generators satisfy). That is, it is the unique group in which every element is a product of some sequence of generators and their inverses, and in which there are no equations between group elements that do not follow in a trivial way from the equations gg⁻¹ describing the relation between a generator and its inverse. The elements of a free group may be described as all of the possible reduced words; these are strings of generators and their inverses, in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation.

The Nielsen–Schreier theorem states that if G is a subgroup of a free group, then G is itself isomorphic to a free group. That is, there exists a subset S of elements of G such that every element in G is a product of members of S and their inverses, and such that S satisfies no nontrivial relations.

The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group on n generators, and H is a subgroup of finite index e, then H is free of rank

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