(adsbygoogle = window.adsbygoogle || []).push({}); 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Group homomorphism

**Physics Forums | Science Articles, Homework Help, Discussion**