My textbook asks:

Choose a homomorphism with . Then . What else can we say about ? Nothing significant, as far as I can tell.Write down the formulas for all homomorphisms from into .

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

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

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

Thanks!