Abstract Algebra Homomorphism Proof

[The_Iceflash]

Homework Statement

Let G and H be two groups. If f: G \rightarrow H is a homomorphism, a \in G and b = f(a). If ord(a) = n, ord(b) = m, then n is a multiple of m. (Let e_{1} be the identity of G and e_{2} be the identity of H)

I have to prove that n is a multiple of m.

Homework Equations

N/A

The Attempt at a Solution

I know aⁿ = ?
and that bⁿ= ?
then I conclude from what I get that n and m are multiples.

Can I have some help with this proof? Thanks.

 

[The Chaz]

a^n = e (in G)
b^m = e (in H)...

 

[Tedjn]

A homomorphism has what property?

 

[gauss^2]

Consider these ideas:

(1) Why is n >= m ?
(2) With n >= m established, you can write n = q m + r, where 0 <= r < m and q > 0.