If is a homomorphism then it is completely determined by since generates the group .Write down the formulas for all homomorphisms from into .
If then .
But if is is homomorphism it means the order of must be dividsibly by order of . The order of is six. Therefore, the order of in must be have order . It cannot have order since and so only two possibilites remain, that the order of must be . For order we get , for order we get have order .
The the only possibily homomorphism are:
Now check that these are indeed all homomorphisms.