Example 3

Let X be a set and Γ a group. Denote by $X\times \Gamma \times X$ the graph whose set of vertices is X, whose set of arrows is $X\times \Gamma \times X$, the origin and end mappings being respectively pr$_1$ and pr$_3$. Endowed with the composition law $(x, \gamma , x')\cdot (x', \gamma ', x'') = (x, \gamma \gamma ', x'')$, it is a groupoid. It is moreover the groupoid deduced by transport of structure from the groupoid $(X\times X)\times \Gamma$, product of the groupoid of pairs $X\times X$ and of the groupoid associated with the group Γ (II, p. 163, example 1), by means of the bijection $(x, x', \gamma )\mapsto (x, \gamma , x')$ of $X\times X\times \Gamma$ onto $X\times \Gamma \times X$.

Let $m: X\times \Gamma \rightarrow X$ be a right operation of the group Γ on X. The graph F of the mapping $m$, that is to say the set of triples $(x, \gamma , x')\in X\times \Gamma \times X$ such that $x'=x\gamma$, is the set of arrows of a unique subgroupoid G of $X\times \Gamma \times X$. The orbits of this groupoid are the orbits of Γ in X; if $x\in X$, the isotropy group of G at $x$ is the set of $(x, \gamma , x)$, where $\gamma$ runs through the stabilizer of $x$ in Γ.

Conversely, every subgroupoid G of the groupoid $X\times \Gamma \times X$ such that the mapping (pr$_1$, pr$_2$) of Fl(G) into $X\times \Gamma$ is a bijection is of this form.