Remark

(1) Let G be a group, H a subgroup of G and $\phi : G \to \mathscr{S}_{G/H}$ the homomorphism corresponding to the operation of G on G/H. The kernel of $\phi$ is the intersection of the conjugates of H (no. 2, Proposition 2). It is also the largest normal subgroup contained in H (no. 3). In particular, G operates faithfully on G/H if and only if the intersection of the conjugates of H reduces to $\{e\}$.

(2) Let G be a group and H and K subgroups such that H is a normal subgroup of K. Then K/H operates on the G-set G/H on the right and the canonical mapping of G/H onto G/K defines on passing to the quotient a G-set isomorphism $(G/H)/(K/H) \to G/K$ (cf. no. 4).