Order of Subgroup Product/Lemma

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Lemma for Order of Subgroup Product

Let $h_1, h_2 \in H$.

Then:

$h_1 K = h_2 K$

if and only if:

$h_1$ and $h_2$ are in the same left coset of $H \cap K$.


Proof

Let $h_1, h_2 \in H$.

Then:

\(\ds h_1 K\) \(=\) \(\ds h_2 K\)
\(\ds \leadstoandfrom \ \ \) \(\ds h_1^{-1} h_2\) \(\in\) \(\ds K\) Left Cosets are Equal iff Product with Inverse in Subgroup
\(\ds \leadstoandfrom \ \ \) \(\ds h_1^{-1} h_2\) \(\in\) \(\ds H \cap K\) Definition of Set Intersection
\(\ds \leadstoandfrom \ \ \) \(\ds h_1 \paren {H \cap K}\) \(=\) \(\ds h_2 \paren {H \cap K}\) Left Cosets are Equal iff Product with Inverse in Subgroup

$\blacksquare$


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