Is Every Group with Squared Elements Equal to the Identity Element Abelian?

[e(ho0n3]

[SOLVED] An Abelian Group Problem

Homework Statement
Prove: If G is a group where the square of every element equals the identity element, then G is Abelian.

The attempt at a solution
I've been able to prove is that a^-1 = a and that (ab)^-1 = ba where a and b are in G. Everything else I've done leads into a dead-end. Any tips?

 

[HallsofIvy]

Okay, you know that a^-1= a for every element of the group (including ab) and you know that (ab)^-1= ba. What do those two together tell you?

 

[e(ho0n3]

I see now! Boy do I feel stupid. Thanks.

 

[HallsofIvy]

Hey, feeling stupid is the beginning of learning!