I need to find the kernel and image of the following group homomorphism:
$$\varphi:\mathbb Z\to S_7, n\mapsto \sigma^n,$$
where $\sigma=\begin{pmatrix}1&2&3&4&5&6&7\\3&4&5&6&7&2&1\end{pmatrix}$.
I already proved that φ is homomorphism. I don't really know where to start with finding them and could use some help in what to use.
The image is just all the elements you can actually reach using your map $\varphi$. Since those are all of the form $\sigma^n$, and $\sigma$ has finite order (given that it's an element of a finite group), you can just list all of them. For concise notation, you can find the smallest positive $n$ for which $\sigma^n=\mathrm{id}$ and write $\operatorname{Im}(\varphi)=\{\sigma^0,\dots,\sigma^{n-1}\}$. This is because starting at $n$, the permutations will repeat.
For the kernel, the work you did for the image makes it so you're almost done. The kernel is made up of those $n\in\mathbb Z$ sent to $\mathrm{id}$. That's exactly the integer multiples of the $n$ from before (the order of $\sigma$). In fact, a nice definition of the order of an element $g\in G$ of any group $G$ is the generator of the kernel of the homomorphism $\mathbb Z\to G, 1\mapsto g$.