Define what it means for a group to be cyclic, and for a group to be abelian. Show that every cyclic group is abelian, and give an example to show that the converse is false.

Show that a group homomorphism from the cyclic group $C_{n}$ of order $n$ to a group $G$ determines, and is determined by, an element $g$ of $G$ such that $g^{n}=1$.

Hence list all group homomorphisms from $C_{4}$ to the symmetric group $S_{4}$.