Let (G, * ) be a group. We are looking for an e − 1 ∈ 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.