Here is a problem from some russian book of algebra:

$ \displaystyle \varphi(x)=y\leftrightarrow\varphi(y)=x$ and I know $ \displaystyle \varphi(e)=e.$ I can see from this that $ \displaystyle G$ is a group of odd order. How I prove commutativity? Do you think I can prove first that $ \displaystyle \varphi(a)=a^{-1}$?Quote: