Definition 7

Let $G$ be a group. An operation of $G$ on a set $E$ is called simply transitive if there exists an element $x$ of $E$ such that the orbital mapping defined by $x$ is a bijection. A set $E$, together with a simply transitive left action of $G$ on $E$, is called a left homogeneous principal $G$-set (or left homogeneous principal set under $G$).

It amounts to the same to say that $G$ operates freely and transitively on $E$, or also that there exists an element $x \in E$ such that the orbital mapping defined by $x$ is an isomorphism of the $G$-set $G$ (where $G$ operates by left translation) onto $E$; or also that the two following conditions are satisfied:

(i) $E$ is non-empty. (ii) for all elements $x$ and $y$ of $E$, there exists one and only one element $\alpha \in G$ such that $\alpha x = y$.

Condition (ii) is also equivalent to the following condition:

(iii) the mapping $(\alpha, x) \mapsto (\alpha x, x)$ is a bijection of $G \times E$ onto $E \times E$.

We leave to the reader the task of defining right homogeneous principal G-sets.