Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 6 / b / iii

Every left coset $kH$ with $k\in K$ is also a left coset of $H$ in $G$. Moreover equality of two such cosets means the same thing in $K$ and in $G$:Thus the map from the set of left cosets of $H$ in $K$ to the set of left cosets of $H$ in $G$, taking $kH$ to itself, is injective. Consequently the index of a subgroup satisfiesThese are sets of cosets; no normal subgroup assumption is needed.