# Intersection of Left Cosets of Subgroups is Left Coset of Intersection

## Theorem

Let $G$ be a group.

Let $H, K \le G$ be subgroups of $G$.

Let $a, b \in G$.

Let:

$a H \cap b K \ne \O$

where $a H$ denotes the left coset of $H$ by $a$.

Then $a H \cap b K$ is a left coset of $H \cap K$.

## Proof

Let $x \in a H \cap b K$.

Then:

 $\ds x$ $\in$ $\ds a H$ $\ds \leadsto \ \$ $\ds x H$ $=$ $\ds a H$ Left Cosets are Equal iff Element in Other Left Coset

and similarly:

 $\ds x$ $\in$ $\ds b K$ $\ds \leadsto \ \$ $\ds x K$ $=$ $\ds b K$ Left Cosets are Equal iff Element in Other Left Coset

Hence:

 $\ds a H \cap b K$ $=$ $\ds x H \cap x K$ $\ds$ $=$ $\ds x \paren {H \cap K}$ Corollary to Product of Subset with Intersection

Hence the result by definition of left coset.

$\blacksquare$