This is pretty straight forward for groups of 3 and 4 elements. Is there a better approach to groups with 5 elements?
I think this will work.
Let .
If then .
If then .
So suppose that the two aren't equal and inverses of each other. Then is an element that is not equal to or . This goes for as well.
Now suppose that . Then we have
which means that If and then
which is a contradiction. The case for is similar.