MHB Group Isomorphism: Proving G Is an Odd, Ablian Group

[Andrei1]

Here is a problem from some russian book of algebra:

Suppose $$G$$ is a finite group. An automorphism $$\varphi$$ "operates" on this group. This automorphism satisfies the following two conditions: 1) $$\varphi^2=e_G$$; 2) if $$a\not= e$$, then $$\varphi(a)\not= a.$$ Prove that $$G$$ is an abelian odd group.

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

 

[johng1]

Hi,
This is a "standard" exercise. Here is a link to a proof: If a group has an automorphism which is fixed point free of order 2, then it is abelian | Project Crazy Project

 

[Deveno]

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

as johng's post shows, the answer is yes.