Proving the Commutativity of a Group with Abstract Algebra

[stihl29]

Let G be a group. Show (xy)$$^{-1}$$ = x$$^{-1}$$y$$^{-1}$$ for all x, g $$\in$$ G if and only if G is abelian.

Homework Equations

The Attempt at a Solution

 

[gomunkul51]

I think you're done :)

 

[Mark44]

Let G be a group. Show (xy)$$^{-1}$$ = x$$^{-1}$$y$$^{-1}$$ for all x, g $$\in$$ G if and only if G is abelian.

Homework Equations

The Attempt at a Solution

What have you tried?

 

[hunt_mat]

The element $$(xy)^{-1}$$ is the inverse element of $$xy$$ and therefore $$(xy)^{-1}xy=e$$ where e is the identity, so...