Inverse of Automorphism is an Automorphism

[BrianMath]

Homework Statement

Verify that the inverse of an automorphism is an automorphism.

Homework Equations

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/" , and this came up as an exercise in lecture 3. I was wondering if I did this problem correctly.

 

[micromass]

Homework Statement

Verify that the inverse of an automorphism is an automorphism.

Homework Equations

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/" , and this came up as an exercise in lecture 3. I was wondering if I did this problem correctly.

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