Example

(1) Let G be a group and consider the operation of G on itself by inner automorphisms. Two elements of G which are conjugate under this operation are called conjugate under inner automorphisms or simply conjugate. The orbits are called conjugacy classes. Similarly, two subsets H and H' of G are called conjugate if there exists an element $\alpha \in G$ such that $H' = \alpha . H . \alpha^{-1}$, that is if they are conjugate under the extension to $\mathscr{P}(G)$ of the operation of G on itself by inner automorphisms.

(2) *In the space $\mathbf{R}^n$, the orbit of a point x under the action of the orthogonal group $\mathbf{O}(n, \mathbf{R})$ is the Euclidean sphere of radius $\|x\|_*$.

The stabilizers of two conjugate elements of E are conjugate subgroups of G (no. 2, Proposition 2).

The quotient set of E under the relation of conjugation is the set of orbits of E; it is sometimes denoted by E/G or G\|E. (Sometimes the notation E/G is reserved for the case where E is a right G-set and the notation G\|E for the case where E is a left G-set.)

Let G be a group operating on a set E on the right. Let H be a normal subgroup of G. The group G operates on E/H on the right, the corresponding law of right action being $(xH, g) \mapsto xHg = xgH$; under this operation, H operates trivially, whence a right operation of G/H on E/H. Let $\phi$ be the canonical mapping of E/H onto E/G; the inverse images under $\phi$ of the points of E/G are the orbits of G (or of G/H) in E/H. Hence $\phi$ defines when passing to the quotient a bijection, called canonical, of $(E/H)/G = (E/H)/(G/H)$ onto E/G.

Let G (resp. H) be a group operating on a set E on the left (resp. right). Suppose that the actions of G and H on E commute, that is

The action of H on E is also a left operation of the opposite group $H^0$ to H. It then follows from § 4, no. 9, Proposition 12 that the mapping which associates with the element $(g, h) \in G \times H^0$ the mapping $x \mapsto g . x . h$ of E into itself is a left operation of $G \times H^0$ on E. The orbit of an element $x \in E$ under this operation is the set $GxH$. The set of these orbits is denoted by $G\|E/H$. On the other hand, the operation of G (resp. H) is compatible with the relation of conjugation with respect to the operation of H (resp. G) and the set of orbits $G\|(E/H)$ (resp. $(G\|E)/H$) is identified with $G\|E/H$: in the diagram

(where $\alpha, \beta, \gamma, \delta, \varepsilon$ denote the canonical mappings of taking quotients), $\gamma \circ \alpha = \delta \circ \beta = \varepsilon$.

Let G be a group and H a subgroup of G. Consider the right operation of H on G by right translation (no. 1, Example 2). The set of orbits $G/H$ is the set of left cosets modulo H; note that G operates on $G/H$ on the left by the law $(g, xH) \mapsto gxH$ (cf. no. 5). Similarly, the set of right cosets modulo H is the set $H\backslash G$ of orbits of the left operation of H on G by left translation. If K is a subgroup of G containing H and $\Gamma$ is a left (resp. right) coset modulo H, then $\Gamma K$ (resp. $K\Gamma$) is a left (resp. right) coset modulo K. The mapping $\Gamma \mapsto \Gamma K$ (resp. $\Gamma \mapsto K\Gamma$) is called the canonical mapping of $G/H$ into $G/K$ (resp. of $H\backslash G$ into $K\backslash G$). It is surjective.

Let G be a group and H and K two subgroups of G. Let H operate on G on the left by left translation and K on the right by right translation; these two operations commute, which allows us to consider the set $H\backslash G/K$. The elements of $H\backslash G/K$ are called the double cosets of G modulo H and K. When $K = H$, we simply say double cosets modulo H. For the canonical mapping of $G/H$ onto $H\backslash G/H$ to be a bijection, it is necessary and sufficient that H be a normal subgroup of G.