Given a finitely generate free group $G$ with generators $s_1, \dots, s_n$. Let $(b_1, \dots, b_n)$ denote a basis of this free group. A elementary transformation of this basis creates a new basis $(b'_1, \dots, b'_n)$ by one of the following to operations:
- For some $1 \leq i \leq n$ replace $b_i$ with $b_i^{-1}$.
- For some $1 \leq i, j \leq n$ with $i \neq j$ replace $b_i$ by either $b_j \cdot b_i$, $b_j^{-1} \cdot b_i$, $b_i \cdot b_j$ or $b_i \cdot b_j^{-1}$, i.e. multiply $b_i$ from the left or the right by $b_j$ or its inverse.
- Exchange $b_i$ and $b_j$.
Two bases of $G$ are called equivalent if one can get the second in a finite number of elementary transformations from the first.
Question: Are all bases equivalent? If not, how many equivalence classes do exist? (finite, infinite?)
Examples: $(s_1, s_2)$ is equivalent to $(s_1^{-1}, s_1^{-1} s_2 s_1)$.