Group Isomorphism: Proving G Is an Odd, Ablian Group

[Andrei1]

Here is a problem from some russian book of algebra:

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

\(\displaystyle \varphi(x)=y\leftrightarrow\varphi(y)=x\) 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 \(\displaystyle \varphi(a)=a^{-1}\)?

 

[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:\(\displaystyle \varphi(x)=y\leftrightarrow\varphi(y)=x\) 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 \(\displaystyle \varphi(a)=a^{-1}\)?

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