Exercise 3.2

Let \((G, *)\) be a group. We are looking for an \(e^{-1} \in G\) such that \(e * e^{-1} = e^{-1} * e = e\) (remember that inverses are unique, so there must be exactly one such \(e^{-1}\)).

By the monoid identity law (remember all groups are monoids), we have that \(e * e = e\), so the inverse of the identity must be the identity itself.