Let $G$ and $H$ be two groups and let $f:G\longrightarrow H$ be an isomorphism.
Are the two following properties true?
1) If $g$ is a generator of $G$ then $f(g)$ is a generator of $H$ and more generally if $\langle g_1,\dotsc,g_n | R_G \rangle $ is a presentation of $G$, $R_G$ being a set of relations, then $\langle f(g_1),\dotsc,f(g_n) | f(R_G) \rangle $ is a presentation of $H$.
2) If $h$ is a generator of $H$, then $f^{-1}(h)$ is a generator of $G$.