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Math Help - Homomorphisms in cyclic group Z8.

  1. #1
    Junior Member platinumpimp68plus1's Avatar
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    Homomorphisms in cyclic group Z8.

    So I'm asked to find the number of automorphisms (homomorphism mapping it to itself) in the cyclic group Z8 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?
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  2. #2
    Super Member Gamma's Avatar
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    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 \mathbb{Z}_8 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 \phi(n) which gives the number of elements relatively prime to n. So you can check to see that your answer is indeed correct.
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  3. #3
    Junior Member platinumpimp68plus1's Avatar
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    Ahhh allright, that makes much more sense. I figured I was confused about the details and that was messing me up. Thank you!
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