1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Group homomorphism
  1. Group Homomorphism (Replies: 1)

  2. Homomorphism of groups (Replies: 2)

  3. Group Homomorphism? (Replies: 3)