TITLE: The free group $Z*Z$ is isomorphic to which group?
QUESTION [3 upvotes]: The free group $\mathbb{Z}*\mathbb{Z}$ is basically an infinite non abelian gorup with each of its elements having infinite order except identity. I can't think of any other group which can be thought as an isomorphic group to this free group.
And can someone please define any homomorphism from $\mathbb{Z}*\mathbb{Z}$ to $\mathbb{Z}$?

REPLY [2 votes]: An example is the commutator subgroup of the modular group
is a free group of rank 2 isomorphic to $F:=\mathbb{Z}*\mathbb{Z}$.
Given two generators $x$ and $y$ of $F$, any integers $n$ and $m$
and the generator $z$ of $\mathbb{Z}$,
then the general homomorphism $f:F \to \mathbb{Z}$ given by $f(x)=z^n,\;f(y)=z^m$
is uniquely well defined since $x$ and $y$ are generators of
the free group $F$.
{"set_name": "stack_exchange", "score": 3, "question_id": 2384562}