I have shown that there are distinct elements a and b st $\displaystyle ab\neq ba.$ I have also shown that there is a unique identity element.

Since G is a group, there a and b must have inverses. I have shown that $\displaystyle a^{-1}\neq e, b^{-1}\neq e, a^{-1} \neq b$ and $\displaystyle b^{-1}\neq a.$ What I can not find out is how to show that $\displaystyle a^{-1}\neq a$ and $\displaystyle b^{-1}\neq b.$ Does anyone have any ideas?