Here's what I understand:
Given a group $G$, a principal homogeneous space for $G$ consists of a set $X$ together with a free and transitive action of $G$ on $X$. Free in this context means that for all $x \in X$, the function $g \mapsto gx$ is injective. Transitive means that the aforementioned function is surjective.
Principal homogeneous spaces for $G$ form a group, at least under certain assumptions. In algebraic geometry this is called Weil–Châtelet group.
Perhaps it's obvious, but I can't see how such a group could be defined, and finding an elementary account of such things has proved difficult.
Question.
How is this group defined?
What's the minimum assumptions on (and/or structure on $G$) necessary for this group to be defined?