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Math Help - Abelian group homomorphism

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    Abelian group homomorphism

    Let G be an abelian group and let n be any positive integer. Show that the function c:G--->G defined by c(x)=x^n is a homomorphism.

    I started out by thinking of the definition of a homomorphism. We have a homomorphism if c(ab)=c(a)c(b).
    So c(a)c(b)
    a^nb^n
    since G is abelian group, a^nb^n=(ab)^n=c(ab)?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kathrynmath View Post
    Let G be an abelian group and let n be any positive integer. Show that the function c:G--->G defined by c(x)=x^n is a homomorphism.

    I started out by thinking of the definition of a homomorphism. We have a homomorphism if c(ab)=c(a)c(b).
    So c(a)c(b)
    a^nb^n
    since G is abelian group, a^nb^n=(ab)^n=c(ab)?
    Right, I think. You can show by induction that since G is abelian that (ab)^{\text{anything}}=a^{\text{anything}}b^{\text  {anything}} which is precisely what it means for x^{\text{anything}} to be a homo.
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