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Group isomorphisms

Andrei

Member
Jan 18, 2013
36
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
Feb 15, 2012
1,967
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.