# Group isomorphisms

#### Andrei

##### Member
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 $$\displaystyle \varphi(a)=a^{-1}$$?

#### Deveno

##### Well-known member
MHB Math Scholar
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.