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...