Drawing from another question that was recently posted:
If $G$ is a group of order 42 and we have $\varphi:G\to G$ with $\varphi(G)$ isomorphic to $\mathbb{Z}/21\mathbb{Z}$.
How can we show that the kernel of $\varphi$ is a subset of $Z(G)$?
I thought of using the First Isomorphism Theorem, but that would only help us to show that the kernel is a normal subgroup of $G$. Any thoughts?