Proving Existence of Element of Order n in Finite Group via Homomorphism

[strangequark]

Homework Statement

Suppose that \phi 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.

Homework Equations

for a homomorphism,
1) \phi(a*b)=\phi(a)*\phi(b)
2) \phi(a^{n})=(\phi(a))^{n}
3) \phi(e_{G})=e_{G'}

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, g^{8} is obviously an element of the kernal of G, and the homomorphism is not the trivial map because g^{n} for 0<n<8 is not the identity in G and doesn't map to the identity in G'. Basically, I'm seeing that g^{8} maps to the identity in G', but I don't understand why this implies that g^{8}=e...

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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