Proof that elements of a free generating set have infinite order.

Question

I'm trying to show that elements of a free generating set $S$ have infinite order straight from the definition of a free group being generated by $S$.

The definition I'm using is that a group $F$ is freely generated by a subset $S$ of $F$ if for any group $\Gamma$ and any map $\phi : S\rightarrow \Gamma$ there is a unique homomorphism from $F$ to $\Gamma$ extending $\phi$.

Answer 1

Let $F$ be free with generating set $S$ and $s\in S$. Then the map $\phi\colon S\to\mathbb Z$, $x\mapsto 1$ extends to a homomorphism $F\to\mathbb Z$ that maps $s^n\mapsto n\ne0$ if $n\ne0$.