Uniqueness of the Identity Element and of Inverses in a Group

Let $(G,\ast)$ be a group. Then the following hold.

1. There is exactly one identity element of $(G,\ast)$. It is denoted by $e_G$, or simply by $e$ when the group is clear from the context.
2. For every $a\in G$ there is exactly one element $b\in G$ satisfying

$$a\ast b=e_G\qquad\text{and}\qquad b\ast a=e_G.$$

It is called the inverse of $a$ and is denoted by $a^{-1}$. 3. The set $G$ is nonempty.

Accordingly, we use throughout the notation $e_G$ for the identity element of a group and $a^{-1}$ for the inverse of an element $a$, in addition to the abbreviation $ab$ for $a\ast b$ introduced in Group and Abelian Group.