1. Here is a problem from some russian book of algebra:
Quote:
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 $\displaystyle \varphi(a)=a^{-1}$?

2. Hi,
This is a "standard" exercise. Here is a link to a proof:

3. Originally Posted by Andrei
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.

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