Let $G$ be a group, and $g$ be an element of $G$, $ \ $ then is $ \ (g^{-1})^{-1}=g \ $ always true?
I don't see any reason why this shouldn't be the case. After all, if the inverse of $g$ is $g^{-1}$, then the inverse of $g^{-1}$ is $g$, by definition. So it would make sense that indeed $(g^{-1})^{-1}=g$.
What is making me hesitant nonetheless is that my textbook has a tendency of always explicitly stating $(g^{-1})^{-1} \ $ instead of $g$ $ $ when a question is about the inverse of $g^{-1}$, $ $ as if the two were not equivalent.
Am I missing something about the notion of inverse elements of a group, or this is pure semantics?
Thanks.
One group theory axiom states that for each $g$ there exists an $h$ satisfying $gh=hg=e$. We can easily prove such an $h$ is unique, so it is denoted $g^{-1}$. Since $g$ is the unique $h$ for which $g^{-1}h=hg^{-1}=e$, $g$ is the inverse of $g^{-1}$ i.e. is $(g^{-1})^{-1}$.