Definition 1. Let $G$ be a group and $H$ be a subgroup. The index of $H$ in $G$ is
\[\lvert G / H \rvert,\]the cardinality of the left coset quotient of $H$.
Proposition 1. For any three groups $A \leq B \leq C$,
\[\lvert C / A\rvert = \lvert C / B \rvert \cdot \lvert B / A \rvert.\]