Let $F$ be a free group of rank at least two and $\alpha,\beta$ be two elements in $F$. Let $N(\alpha)$ (resp. $N(\beta)$) be the normal subgroup generated by $\alpha$ (resp. $\beta$); i.e., $N(\alpha)$ is the minimal normal subgroup of $F$ containing $\alpha$.
Question: if $N(\alpha)=N(\beta)$, should $\alpha$ be a conjugate of $\beta$ or a conjugate of $\beta^{-1}$?
One motivation: Two groups each defined by one generator might be isomorphic but the two generators might look "quite different". For example, $\langle a,b: a^2b^2\rangle$ is isomorphic to $\langle a,b:aba^{-1}b\rangle$, as they are different representations the fundamental group of Klein bottle. The two generators are not a pair of conjugates, but the two generators don't generate the same normal subgroups either.
The answer is positive. If two elements have the same normal closure in a free group, then one is conjugate to the other, or conjugate to the inverse of the other. This is a theorem of Magnus, which can be found in Lyndon and Schupp's book.