# Homomorphism abelian help

1. Jul 13, 2007

### ldelong

1. The problem statement, all variables and given/known data
Prove that Group G is abelian iff the function f:G to G given by f(a)=a^-1 is a homomorphism

2. Relevant equations

3. The attempt at a solution

Group G must be communitative for it to be abelian I have no idea where to start to begin to prove this I know that homomorphic means that complete the operation first on a in the group gets mapped to an element that the operation is completed 2nd. Help... I really need a jumping point.

2. Jul 13, 2007

### StatusX

You can start by writing out some relevant definitions and basic facts along with the facts you're given. Often this is also where you finish. In this case, remember that the condition for f to be a homomorphism is f(ab)=f(a)f(b). Now use the definition of f.

Last edited: Jul 13, 2007
3. Jul 14, 2007

### matt grime

When I see things like that I worry. Commutative and abelian are synonyms.

4. Jul 14, 2007

### CompuChip

Actually there's not much to prove. It is, as StatusX already said, a matter of writing out the definition of homeomorphism and plugging in commutativity of the group and the definition of f at the right points.

So just start writing something down, then tell us where you get stuck.

5. Jul 16, 2007

### ldelong

I got it thanks