Find all group homomorphisms from $\mathbb{Z}$ to $S_5$.
I think to use the first theorem of homomprphism. Since $\mathbb{Z}$ is cyclic, the image of homomorphism is a cyclic subgroup of $S_5$. The orders of elements in $S_5$ are $1,2,3,4,5,6$, and I have to find how many each of them are. They are $1,25,20,30,24,20$, respectively. Let us map $1$ to $x$ where $x$ is any element of $S_5$ of order $3$. Then I think in this way I will get $10$ homomorphisms. Same with other order elements. Is it correct?