Thread: [SOLVED] homomorphism issue on Z_m \to Z_n

My textbook asks:

Write down the formulas for all homomorphisms from into .

Choose a homomorphism with . Then . What else can we say about ? Nothing significant, as far as I can tell.

But my textbook has another story. According to it, there are only

3 different homomorphisms, given by the formulas , , or , defined for all .

First of all, I am not completely certain I understand the notation, here. Is simply another way of writing ? If so, then by algebra of integers modulo, isn't true for all ? If not, then what does the notation mean?

As you can see, I'm a little bit befuddled by this. I would welcome any assistance.

Thanks!

Write down the formulas for all homomorphisms from into .

If is a homomorphism then it is completely determined by since generates the group .
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.