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Homework Help: Group homomorphism

  1. Feb 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that [tex]\phi[/tex] is a homomorphism from a finite group G onto G' and that G' has an element (g') of order n. Prove that G has an element of order n.

    2. Relevant equations
    for a homomorphism,
    1) [tex]\phi(a*b)=\phi(a)*\phi(b)[/tex]
    2) [tex]\phi(a^{n})=(\phi(a))^{n}[/tex]
    3) [tex]\phi(e_{G})=e_{G'}[/tex]

    3. The attempt at a solution

    It is clear to me that G will contain some non-identity element, say g, which is the preimage of g'. By property 2) that I listed above, [tex]g^{8}[/tex] is obviously an element of the kernal of G, and the homomorphism is not the trivial map because [tex]g^{n}[/tex] for 0<n<8 is not the identity in G and doesn't map to the identity in G'. Basically, I'm seeing that [tex]g^{8}[/tex] maps to the identity in G', but I don't understand why this implies that [tex]g^{8}=e[/tex]...

    I would really appreciate a kick in the right direction... thanks
  2. jcsd
  3. Feb 28, 2008 #2
    You're completely right. It doesn't imply g^8 = e. You're going to have to do a bit more work than just finding an element of the preimage of g'. Unfortunately, I'm having trouble coming up with the exact proof off the top of my head, so I can't give you much of a kick in the right direction. Look at some properties of preimages of subgroups maybe?
  4. Feb 28, 2008 #3
    Let T(a)^8=I', then a has to have an order n*8, and (a^n)^8=I
    This is also summarized by some subgroup divisibility theorem; but it's been to long for me to remember.
    For an Image: Imagine a large gear driving a small gear but every tooth of the large gear strikes the same tooth of the small gear on every revolution.
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