**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 [itex]f:G\to G[/itex] be an automorphism. Then, [itex]f(xy)=f(x)f(y)[/itex] [itex]\forall x,y\in G[/itex].

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

I was watching

Harvard's video lectures on Abstract Algebra, and this came up as an exercise in lecture 3. I was wondering if I did this problem correctly.