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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $3$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $9$