Order of Homomorphisms and Finite Elements

[Locoism]

Homework Statement

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

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.

 

[micromass]

First show that

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

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

 

[Locoism]

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

 

[micromass]

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

Yes to both.