Thread: Homomorphisms in cyclic group Z8.

So I'm asked to find the number of automorphisms (homomorphism mapping it to itself) in the cyclic group under addition.

I think this should be 4, to correspont to <1>,<3>,<5>, and <7>, because these are relatively prime to 8 and act as generators for the group, and I know that the homomorphism should map generators to generators.

But I'm kind of unsure as to why this is. I don't know how to prove it/can't find more information about it. Am I right? Any insight?

I think some of your confusion lies in the fact that you do not quite have the right definition of Automorphism. And automorphism is much more than just a homomorphism from G to G. It is actually an isomorphism, which means it is bijective with an inverse which is also an isomorphism. It turns out that since is generated by 1, you are correct in assuming it suffices to see where 1 can be sent and this will determine all automorphisms. Furthermore, you have indeed chosen the correct places to which 1 can be sent. Because it is an isomorphism, elements must be sent to elements with the same order in G. 1 has order 8, and all of the elements you named also have order 8, and thus are generators. So this is actually a complete list of all possible automorphisms.

It is actually true in general that the size of the automorphism group of any cyclic group of size n is given by the Euler Totient function which gives the number of elements relatively prime to n. So you can check to see that your answer is indeed correct.

Ahhh allright, that makes much more sense. I figured I was confused about the details and that was messing me up. Thank you!