Permutation_of_Cosets

Let $G$ be a group and let $H \le G$. Let $\mathbb S$ be the set of all distinct left cosets of $H$ in $G$. Then $(1): \quad$ For any $g \in G$, the mapping $\theta_g: \mathbb S \to \mathbb S$ defined by $\map {\theta_g} {x H} = g x H$ is a permutation of $\mathbb S$. $(2): \quad$ The mapping $\theta$ defined by $\map \theta g = \theta_g$ is a homomorphism from $G$ into the symmetric group on $\mathbb S$. $(3): \quad$ The kernel of $\theta$ is the subgroup $\displaystyle \bigcap_{x \mathop \in G} x H x^{-1}$.