Tower law for indices

Statement

Let $G$ be a group, let $H$ be a subgroup of $G$, and let $K$ be a subgroup of $H$. Then $[G : K] = [G : H] \cdot [H : K]$, where $[\cdot : \cdot]$ is the index.

Remarks

The equation holds as an equation of cardinal numbers; in particular $[G : K]$ is finite if and only if $[G : H]$ and $[H : K]$ are both finite. For example, in $(\mathbb{Z}, +)$ with $H = 2\mathbb{Z}$ and $K = 6\mathbb{Z}$, the indices are $[\mathbb{Z} : 6\mathbb{Z}] = 6 = 2 \cdot 3$.

Explanations

L1 human

Each coset of $H$ in $G$ splits into $[H : K]$ cosets of $K$. So the number of cosets of $K$ is the number of cosets of $H$ times $[H : K]$. It is like a box with several drawers, each with the same number of compartments.