Index of Intersection of Subgroups/Corollary

Theorem
Let $G$ be a group. Let $H$ be a subgroup of $G$.

Let $K$ be a subgroup of finite index of $G$.

Then:


 * $\index H {H \cap K} \le \index G K$

where $\index G K$ denotes the index of $K$ in $G$.

Note that here the symbol $\le$ is being used with its meaning less than or equal to.

Equality holds $H K = \set {h k: h \in H, k \in K} = G$.

Proof
Note that $H \cap K$ is a subgroup of $H$.

From Index of Intersection of Subgroups, we have:


 * $\index G {H \cap K} \le \index G H \index G K$

Setting $G = H$, we have:


 * $\index H {H \cap K} \le \index H H \index H K$