Group homomorphism

strangequark

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

Mystic998

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?

rrogers

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.
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