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Math Help - Proving isomorphism to Zn

  1. #1
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    Question Proving isomorphism to Zn

    I have proven cyclic group to be a group, but now must prove cyclic group order n to be isomorphic to Zn. I realize I must show homomorphism, injection and surjection. But not struggling with initial equivalences. Please help.
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    Re: Proving isomorphism to Zn

    Let a be a generator of your cyclic group G, can you find a function such that f:G \to \mathbb{Z}_n and f(a)=1 such that f is an homomorphism?
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    Re: Proving isomorphism to Zn

    are you suggesting a concrete f like f(x)=x+5 or generalized like f(x)=ix+j.
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    Re: Proving isomorphism to Zn

    Something concrete, in terms of things we know must exist (like generator "a").
    Try a^n
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    Re: Proving isomorphism to Zn

    Quote Originally Posted by MAdone View Post
    are you suggesting a concrete f like f(x)=x+5 or generalized like f(x)=ix+j.
    ????

    Suppose the operation of your group is given by +, if an f as I described existed, what would be the image of ka=a+a+a\hdots a ( k summands) under f, and a is a generator so...
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    Re: Proving isomorphism to Zn

    all elements of the cyclic group
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    Re: Proving isomorphism to Zn

    Quote Originally Posted by Jose27 View Post
    Let a be a generator of your cyclic group G, can you find a function such that f:G \to \mathbb{Z}_n and f(a)=1 such that f is an homomorphism?
    MAdone, this is really the whole of the idea.

    if f:G \to \mathbb{Z}_n is to be a homomorphism, we must have f(a*a) = f(a) + f(a) = 1+1. so there is really only "one" essential way to define f:

    f(a^k) = k. that f is a homomorphism follows immediately from the laws of exponents.

    since G is finite it suffices to show that f is surjective (clearly |G| = |\mathbb{Z}_n| = n), and thus f is an isomorphism.
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