If G is a group in which every element is its own inverse, then G is abelian.
(HA! Too easy. I LOVE proving that things are abelian, and this is super easy. Wanna see? Too late!)
NOTE: I am not saying that if you can't do this you are not smart, but compared to other things in group theory I think this is one of the easiest things to do.
NTS. If G is a group in which every element is its own inverse, and then G is abelian.
We can do the same for b.
Since we can cancel and therefore which shows that G is abelian.
I also finally just got latex for blogger to work and wanted to use it. Also, I just really liked doing this proof. It's fun, and did I mention easy?