Does the inverse element of a group also have to be under the same set of the group?

Question

In other words, lets say there is a group $G$. If there is an element $g\in G$, by definition there has to be an inverse $g^{-1}$. Now, my question is, does $g^{-1}$ have to be an element $\in G$? Sorry if this may sound obvious but I want to make sure as the definition doesn't specifically state this.

Answer 1

Yes, the following is from wikipedia page of Group (mathematics):

For each $a$ in $G$, there exists an element $b$ in $G$, commonly denoted $a^{−1}$ (or $−a$, if the operation is denoted "$+$"), such that $a \cdot b = b \cdot a = e$, where $e$ is the identity element.

Answer 2

Since the group operation is only required to be on $G$, to make sense of $g \cdot g^{-1}$, we must have $g^{-1} \in G$.