---
tag: B2TRERFWEH
kind: statement
role: theorem
name: Magnus's Freiheitssatz
msc: ["20F05", "20E05"]
uses: [68AFPHR4GN, 0KTW26C80J, 0JGEQXQ4AC, 0X1JXT47GA]
about: [68AFPHR4GN]
sources:
  - url: https://en.wikipedia.org/wiki/Freiheitssatz
same_as:
  wikidata: Q5501402
  wikipedia: Freiheitssatz
---

## Statement

Let $F$ be the [[free group|0KTW26C80J]] on a set $X$, and let $r \in F$ be a cyclically reduced word.
Let $G = \langle X \mid r \rangle$ be the [[one-relator group|68AFPHR4GN]] it defines.
Let $Y$ be a subset of $X$ that does not contain some generator that occurs in $r$.
Then the [[subgroup|0X1JXT47GA]] of $G$ generated by the images of the elements of $Y$ is free, and these images form a free basis of it.

## Remarks

Magnus proved the theorem by induction on the length of $r$.
After a change of generators, $G$ is written as an [[HNN extension|YEYERRY37K]] or an amalgamated product of one-relator groups with shorter relators, and the induction applies to them.
The same method shows that the [[word problem|PBGTTAVMMT]] of one-relator groups is solvable.

Example: in $\Z^2 = \langle a, b \mid a b a^{-1} b^{-1} \rangle$, the theorem says that $a$ generates an infinite cyclic subgroup, a free group of rank $1$.
In the [[Baumslag–Solitar group|429JJ5ZNBY]] $\langle a, b \mid b a b^{-1} a^{-2} \rangle$, both $a$ and $b$ have infinite order.

The word $r$ must be cyclically reduced.
For $r = b a b^{-1}$, which contains $b$, the group $\langle a, b \mid b a b^{-1} \rangle$ is infinite cyclic generated by $b$, and $a = 1$ in it.
So $\set{a}$ omits $b$, but $a$ does not generate a free group of rank $1$.
The cyclic reduction of $r$ is $a$, which does not contain $b$.

## Explanations

### L1 human

Freiheitssatz is German for "freeness theorem".
A one-relator group is a free group in which one rule is imposed.
The theorem says that the rule cannot affect any set of generators that leaves out a letter of the rule.
Those generators stay as free as they were before.

### L2 human

The theorem is the starting point of the theory of one-relator groups.
Magnus's method of writing a one-relator group as an HNN extension of a group with a shorter relator is the main tool of the area.
Suppose that $r$ is cyclically reduced and that every generator in $X$ occurs in $r$.
Then every proper subset of $X$ is a free basis of a free subgroup of $G$.