Compute $\ker(\phi)$ for $\phi:\mathbb{Z}\to\mathbb{Z}\_7$ such that $\phi(1)=4$

My answer say that![enter image description here](https://i.stack.imgur.com/2yPa0.png)

I have no idea what they mean by "4 has order 7 in $\mathbb{Z}\_7$"

All I can write is that $\ker(\phi) = \{ m \in \mathbb{Z} : \phi(m) =0\}$ because $0$ is the identity in $\mathbb{Z\_7}$ right? But I donn't know what $\phi$ is, so I don't know how to continue. And I have no idea why theya re breaking up the sum inside $\phi$