---
tag: W97X7WX5G8
kind: statement
role: theorem
name: Nielsen–Schreier theorem
msc: ["20E05", "20E07"]
uses: [0KTW26C80J, 0X1JXT47GA, 0DTXBQPA2S]
about: [0KTW26C80J]
sources:
  - url: https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem
same_as:
  wikidata: Q1259814
  wikipedia: Nielsen–Schreier theorem
---

## Statement

Let $F$ be a [[free group|0KTW26C80J]].
Then every [[subgroup|0X1JXT47GA]] $H$ of $F$ is a free group.
Moreover, if $F$ has finite rank $n$ and $H$ has finite [[index|0DTXBQPA2S]] $e = [F : H]$, then $H$ is free of rank $1 + e(n - 1)$.

## Remarks

A topological proof: $F$ is the [[fundamental group|S1XK3HXHFX]] of a bouquet of circles, a graph.
Then $H$ is the fundamental group of a [[covering space|3XZRRE165N]], which is again a graph.
The fundamental group of a connected graph is free.
For the rank formula, the covering graph of degree $e$ of a bouquet of $n$ circles has $e$ vertices and $en$ edges.
So its Euler characteristic is $e(1 - n)$.
A connected graph with Euler characteristic $\chi$ has free fundamental group of rank $1 - \chi$.
An algebraic proof uses a Schreier transversal and gives explicit free generators, the Schreier generators.
Example: in the free group $F(a, b)$, let $H$ be the kernel of the map to $\Z/2$ sending $a$ and $b$ to $1$.
It has index $2$, so it is free of rank $1 + 2 \cdot 1 = 3$; a free basis is $a^2$, $ab$ and $ab^{-1}$.
So a subgroup of a free group can have larger rank than the group.
The free group of rank $2$ contains free subgroups of every finite rank, and of countably infinite rank, such as its [[commutator subgroup|0ASGSTXSBW]].
The finite index hypothesis is needed in the rank formula: the commutator subgroup of $F(a, b)$ has infinite index and is not finitely generated.

## Explanations

### L1 human

A free group is a group with no relations among its generators other than the obvious cancellations.
The theorem says that every subgroup of such a group is again free.
This is surprising, since subgroups of a free group can be complicated, but they never acquire genuine relations.

### L2 human

The theorem is one of the starting points of combinatorial and geometric group theory.
The covering space proof is the model for [[Bass–Serre theory|6M731X66RD]], where groups acting on trees are studied, and the [[Kurosh subgroup theorem|RY8K5RQJSR]] is the analogue for subgroups of [[free products|TM7QS7P7X9]].
The rank formula is a form of the multiplicativity of the Euler characteristic under finite coverings.