Index of Intersection of Subgroups

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Theorem

Let $G$ be a group.

Let $H, K$ be subgroups of finite index of $G$.


Then:

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

where $\index G H$ denotes the index of $H$ 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 $H K = \set {h k: h \in H, k \in K} = G$.


Corollary

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

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

Then:

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


Proof

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

From Tower Law for Subgroups, we have:

$\index G {H \cap K} = \index G H \index H {H \cap K}$

From Index in Subgroup, also:

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

Combining these results yields the desired inequality.


Again from Index in Subgroup, it follows that:

$\index H {H \cap K} = \index G K$

if and only if $H K = G$.

$\blacksquare$


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