Free groups satisfy IBN

Statement

Free groups satisfy the invariant basis number (IBN) property: If $S$ and $T$ are two sets such that the free group on the set $S$ is isomorphic to the free group on the set $T$, then $S$ and $T$ have the same cardinality.

Facts used

• Free Abelian groups satisfy IBN

Proof

We use the fact that the free Abelian group on a set is the Abelianization of the free group on the set. Thus, if the free group on $S$ is isomorphic to the free group on $T$, so are their Abelianizations, and hence the free Abelian group on $S$ is isomorphic to the free Abelian group on $T$. The problem thus reduces to showing that free Abelian groups satisfy IBN.