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Math Help - Homomorphism and order of elements

  1. #1
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    Homomorphism and order of elements

    Hi,

    I was wondering if someone could help me with this question I have.

    "If f: G--> H is a homomorphism, prove the following. "If the range of f has n elements, then xn is in the kern f for every x in G. ""


    I know that the kern f is a set K that has all elements of G that are carried by f onto the neutral element in H.

    I also know that the range of f is a subgroup of H.

    So what the question is asking to prove is that if we take every element of G and raise it to the power n where n is the number of elements in the range, then all of those xn must be mapped to the neutral element of H. Is that correct?

    Any help is greatly appreciated!

    Thanks!
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Homomorphism and order of elements

    Yes.

    Since | \phi(G) | = n. That means, since  \phi(G) is a subgroup, then every element in that subgroup raised to the nth power is e.
    so  (\phi(x))^n = e for  x \in G. since  (\phi(x))^n = \phi(x^n) = e  x^n \in Ker(\phi)
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  3. #3
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    Re: Homomorphism and order of elements

    Quote Originally Posted by jakncoke View Post
     ... = \phi(x^n) = e  x^n \in Ker(\phi)
    I have another question. How you you get from \phi(x^n) = e x^n \in Ker(\phi)?

    Thank you!
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