Is Group G Abelian if f(a)=a^-1 is a Homomorphism?

[ldelong]

Homework Statement

Prove that Group G is abelian iff the function f:G to G given by f(a)=a^-1 is a homomorphism

Homework Equations

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.

 

[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.

 

[matt grime]

Group G must be communitative for it to be abelian

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

 

[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.

 

[ldelong]

I got it thanks