Let $G$ be a set with an associative operation defined on it. Show that $G$ is group.
For all $g$ and $h$ in $G$, $gx=h$ has a unique solution in $G$.
My attempt:
A set $G$ has left cancellation property by uniqueness of solution of equation.
For any $a\in G$, $\exists e_a \in G$ such that $ae_a=a$.
$aa=(ae_a)a=a(e_aa) \implies e_aa=a$.
For $g \in G$, $\exists !h\in G$ such that $ah=g$.
Thus $ e_ag=e_a(ah)=(e_aa)h=ah=g$.
But I don't know how to make the equation $ge_a=g$ is vaild.