How can I prove the following result?
Let $G$ be a group, $X\subseteq G$ and let $F_a(X)$ be the free group on $X$. Then the subgroup of $G$ generated by $X$ is isomorphic with $F_a(X)$ if and only if any normal word is distinct from identity $e\in G$.
We call a normal word of $X$ an element $x_1^{r_1}...x_n^{r_n}\in G$ where $x_i\in X$, $x_i\neq x_{i+1}$ and $r_i\neq 0$ are integers.
My real question is, what is the problem if some normal word is identity? I don't see it.