Right identity and Right inverse implies a group
Let $(G,*)$ be a binary structure that has the following properties:
1) The binary operation $*$ is associative.
2) There exists an element $e \in G$ such that for all $a \in G$, $e*a=a$ (Existence of left Identity).
3) For all $a \in G$, there exists $b \in G$ such that $b*a=e$ (existence of left inverses)
Prove that $(G,*)$ is a group.
This is how I did it.
I want to check that
$a*(a'*a)=a$ , from here I cant go on, because I only have left inverses as well. So I know something is missing. And what should I do to find the right inverse?