Intersection of Left Cosets of Subgroups is Left Coset of Intersection

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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.

$\blacksquare$


Sources

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