Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an infinite cyclic group. Furthermore, every subgroup of $F_n$ is itself a free group, which would seem to suggest that $C_{F_n}(h)$ is a free group on one generator. Any free group on one generator is necessarily abelian.
So my question is, am I correct in concluding that $C_{F_n}(h)$ is in fact an abelian subgroup of $F_n$ for every $h \in F_n$?