Action on left cosets

Example

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $G/H$ be the set of the left cosets of $H$ in $G$. The map $G \times G/H \to G/H$, $(g, aH) \mapsto gaH$, is an action of $G$ on $G/H$. The map is well defined, because $aH = bH$ implies $gaH = gbH$. The action is transitive, because $(ba^{-1}) \cdot aH = bH$. The stabiliser of the coset $aH$ is the conjugate subgroup $aHa^{-1}$. The kernel of the associated homomorphism $G \to \operatorname{Sym}(G/H)$ is the normal core of $H$, the intersection of all conjugates of $H$. For $H = \{e\}$, this is the action by left multiplication.