Does the identity element of a group have an inverse? I can't seem to find anything on the topic.

All elements of a group have an inverse. This is a requirement in the definition of a group.
For an element $g$ in a group $G$, an inverse of $g$ is an element $b$ such that $gb = e$ where $e$ is the identity in the group. (Since the inverse of an element is unique, we usually denoted the inverse of $g$ $g^{-1}$ or $-g$.)
Note now that $ee =e$, so by definition $e$ is an inverse of itself.
You might be wondering if other elements might be their own inverses. The answer to this is yes. For example in the group $\mathbb{Z}_2$ of order $2$, both elements are their own inverse.
