TITLE: Definition of groups
QUESTION [7 upvotes]: This seems like a very basic question but got me confused. When defining a group we introduce the unit element $e$ which has the following property
$$ge = eg = g \quad \forall g\in G$$
and then the inverse for which we need the unit:
$$gg^{-1} =g^{-1}g = e$$
Is it possible to do it the other way around? Can we construct the unit element from the inverse? I think not but haven't found a convincing argument except that we use the unit in the definition of the inverse. But perhaps it is thinkable to conceive a completely different way.

REPLY [2 votes]: You can replace the axiom of the neutral and the inverse element by the following axiom:
For all $a,b \in G$ each of the following equations has a unique solution.
$$ax=b$$
$$xa=b$$
From this axiom the existence of a neutral element and the inverse elements can be deduced.