Example

(1) Let $E$ be a non-empty set. The group $S_E$ operates transitively on $E$. If $x$ and $y$ are two elements of $E$, the mapping $\tau : E \to E$ such that $\tau(x) = y$, $\tau(y) = x$ and $\tau(z) = z$ for $z \neq x, y$, is a permutation of $E$. Let $a \in E$. The stabilizer of $a$ is identified with $S_F$, where $F = E - \{a\}$. The homogeneous $G$-set $E$ is thus isomorphic to $S_E / S_F$.

(2) Let $E$ be a set of $n$ elements and $(p_i)_{i \in I}$ a finite family of integers $> 0$ such that $\sum_i p_i = n$. Let $X$ be the set of partitions $(F_i)_{i \in I}$ of $E$ such that $\mathrm{Card}(F_i) = p_i$ for all $i$. The group $S_E$ operates transitively on $X$. The stabilizer $H$ of an element $(F_i)_{i \in I}$ of $X$ is canonically isomorphic to $\prod_{i \in I} S_{F_i}$ and is hence of order $\prod_{i \in I} p_i!$. Applying Theorem 1 and § 4, no. 4, Corollary to Proposition 4, a new proof is obtained of the fact that

In particular, take $I = \{1, 2, \ldots, r\}$, $E = \{1, 2, \ldots, n\}$,

for $1 \leq i \leq r$. Let $S$ be the set of $\tau \in S_E$ such that $\tau|_{F_i}$ is increasing for $1 \leq i \leq r$. If $(G_1, \ldots, G_r) \in X$ there exists one and only one $\tau \in S$ which maps $(F_1, \ldots, F_r)$ to $(G_1, \ldots, G_r)$. In other words, each left coset of $S_E$ modulo $H$ meets $S$ in one and only one point.

(3) Let $n$ be an integer $\geq 1$. The orthogonal group $O(n, \mathbf{R})$ operates transitively on the unit sphere $S_{n-1}$ in $\mathbf{R}^n$. The stabilizer of the point $(0, \ldots, 0, 1)$ is identified with the orthogonal group $O(n-1, \mathbf{R})$. The homogeneous $O(n, \mathbf{R})$-set $S_{n-1}$ is thus isomorphic to $O(n, \mathbf{R})/O(n-1, \mathbf{R})$.