$G$ cyclic group of order $n$ and $g \in G$ generator; then $g^k$ is a generator iff $(k,n)=1$. Is this a valid argument?

Question

Can I get the result in the title by simply noting that the map of $\{0,\dots,n-1\}$ in itself, defined by $i \mapsto ki \pmod n$, is bijective if and only if $(k,n)=1$?