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A question on homomorphisms

  1. Dec 19, 2004 #1
    Consider the cyclic group G={a,a^2,a^3,...a^12=u} and its subgroup G`={a^2,a^4,...,a^12}. My book says that the mapping
    a^n ---> a^2n is an homomorphism of G onto G` (this seems true)

    and that X: a^n ---> a^n is homomorphism of G` onto G (this seems to be false to me, a misprint)

    A homomorphism of G` onto G would have
    1)every g` in G` has a unique image g in G (true)
    2)if X(a`)=a and X(b`)=b then X(a` o b`)=X(a x b) with operator o for G` and x for G (true)
    3)every g in G is an image (false) e.g. a^3 in G is not the image of any g` in G`.

    In fact, it would seem to me that you can have a homomorphism of a group onto a subgroup, but you could not have a homomorphism of a subgroup onto a group ever. Is this right, or am I missing something?
  2. jcsd
  3. Dec 19, 2004 #2
    Have you confused homomorphism with isomorphism?
  4. Dec 19, 2004 #3


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    Try looking at infinite groups.
  5. Dec 19, 2004 #4
    I see the problem. It is not a misprint, just poor language. The original text says (paraphrased) a homomorphism of G into H as
    F(G,*g) ---> (H,*h)
    F(g1 *g g2)=F(g1) *h F(g2)

    Then the book reads, "if (iii) every h in H is an image we have a homomorphism of G onto H." This is actually a definition of the term ONTO, that is to say epimorphism, or surjective. But if you are not used to all this, it almost makes it sound like a requirement of homomorphism, which I realize now it is not. This is why it sounded like I was confusing isomorphism with homomorphism, as isomorphisms are injective and surjective, that is one-to-one.

    You all have really helped clear this up. Carrying this misunderstanding forward could have been deadly.
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