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Math Help - Counterexample to homomorphism problem

  1. #1
    Forum Admin topsquark's Avatar
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    Counterexample to homomorphism problem

    Here it is:
    Show that if H is any group and h is an element of H with h^n = e, then there is a unique homomorphism from Z_n = < x > to H such that x \mapsto h.
    I have been able to prove such a homomorphism exists with a cyclic subgroup of H anyway. How do I extend this to they whole group H? For example when I do this using H = D_8 I end up with a "homomorphism" that is a relation, not a function.

    And as far as uniqueness goes the homomorphism maps x to h. Isn't that the only function that will do that? I mean if we have \phi : Z_n \to H: x \mapsto h how could we have a distinct function \psi : Z_n \to H: x \mapsto h? If they both map x to h wouldn't they have to be the same function automatically? I'm missing something here...

    -Dan
    Last edited by topsquark; January 8th 2014 at 08:52 AM.
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  2. #2
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    Re: Counterexample to homomorphism problem

    There is no requirement that the homomorphism be surjective....

    The whole point of the problem is that: o(h)|n. If this were NOT so, we would have hn ≠ e, in which case:

    φ(e) = φ(xn) = φ(x)n = hn ≠ e, so φ is not a homomorphism (which must map identity to identity).

    Homomorphisms are just certain functions between groups, the image of a homomorphism does NOT have to be the entire target group.
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