# Homework Help: Group Homomorphisms and order

1. Oct 11, 2011

### Locoism

1. The problem statement, all variables and given/known data
Let ψ: G→H be a homomorphism and let g ε G have finite order.
a) Show that the order of ψ(g) divides the order of g

3. The attempt at a solution
I'm really lost here, but I'm guessing we can use the fact |ψ(g)| = {e,g...,g|g|-1}
and ψ(g|g|-1) = ψ(g)ψ(g)ψ(g)ψ(g)ψ(g).... (|g|-1 times)
I still have no idea where to start.

2. Oct 11, 2011

### micromass

First show that

$$\psi(g)^{|g|}=e$$

Then use the general fact that if $a^n=e$, then |a| divides n.

3. Oct 11, 2011

### Locoism

Ah ok but I would use eH?
Also, if it is an isomorphism, could I show |ψ(g)| = |g|?

4. Oct 11, 2011

Yes to both.