# Inverse of Automorphism is an Automorphism

1. Aug 7, 2011

### BrianMath

1. The problem statement, all variables and given/known data
Verify that the inverse of an automorphism is an automorphism.

2. Relevant equations

3. The attempt at a solution

Let $f:G\to G$ be an automorphism. Then, $f(xy)=f(x)f(y)$ $\forall x,y\in G$.
Then, we define the inverse $f^{-1}:G\to G$ by $f^{-1}(f(x)) = f(f^{-1}(x)) = x$ $\;\;\forall x\in G$. We get $f^{-1}(f(x)f(y))=f^{-1}(f(xy))=xy=f^{-1}(f(x))f^{-1}(f(y))$. Since $f^{-1}(f(x)f(y))=f^{-1}(f(x))f^{-1}(f(y))$, $f^{-1}$ is an automorphism.

I was watching http://www.extension.harvard.edu/openlearning/math222/" [Broken], and this came up as an exercise in lecture 3. I was wondering if I did this problem correctly.

Last edited by a moderator: May 5, 2017
2. Aug 7, 2011

### micromass

Seems good!!
As an exercise however, it would be benificial if you indicated where exactly you use injectivity and surjectivity...

Last edited by a moderator: May 5, 2017