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 P: 26 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 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.