- #1

strangequark

- 38

- 0

## Homework Statement

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.

## Homework 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]

## 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