Exercise 2, § 4

(a) An associative law $(x, y) \mapsto xy$ on a set E is a group law if there exists $e \in E$ such that, for all $x \in E$, $ex = x$ and if, for all $x \in E$, there exists $x'$ such that $x'x = e$ (show that $xx' = e$ by considering the composition $x'xx'$; deduce that $e$ is the identity element).

(b) Show that the same is true if, for all $x \in E$, the left translation $\gamma_x$ is a mapping of E onto E and if there exists some $a \in E$ such that the right translation $\delta_a$ is a mapping of E onto E.