Index of Intersection of Subgroups/Corollary

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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 if and only if $H K = \set {h k: h \in H, k \in K} = G$.


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:

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



$\blacksquare$


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