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Homomorphism from Z into a nonabelian group

  1. Feb 7, 2009 #1

    quasar987

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    What would be an example of a (nontrivial) homomorphism from Z into a nonabelian group??

    More specifically, I am looking for two homomorphisms f, g: Z-->W such that for some n,m in Z f(n)g(m) does not commute.
     
    Last edited: Feb 7, 2009
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  3. Feb 7, 2009 #2

    quasar987

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    Ah! I found out that the smallest nonabelian group is the diedral group of order 6, D_3. And from its multiplication table, we see that embedded in it is a copy of Z_2 and a copy of Z_3 that contain elements that do not commute.
     
  4. Feb 7, 2009 #3

    StatusX

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    A homomorphism from Z into any group G is completely specified by picking f(1), because if f(1)=g, then f(n)=gn. And you can pick f(1) to be any element of G that you want, basically because Z is a free group.
     
  5. Feb 8, 2009 #4

    quasar987

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    Thanks for that remark StatusX.
     
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