Order of Subgroup Product/Proof 2

Lemma
We have that $H K$ is the union of all left cosets $h K$ with $h \in H$:
 * $\ds H K = \bigcup_{h \mathop \in H} h K$

From Left Coset Space forms Partition, unequal $h K$ are disjoint.

From Cosets are Equivalent, each $h K$ contains $\order K$ elements.

From the Lemma, the number of different such left cosets is:
 * $\index H {H \cap K}$

where $\index H {H \cap K}$ denotes the index of $H \cap K$ in $H$.

First, let $\order H < + \infty$.

Then, from Lagrange's Theorem:

$\index H {H \cap K} = \dfrac {\order H} {\order {H \cap K} }$

Hence:
 * $\order {H K} = \dfrac {\order H \order K} {\order {H \cap K} }$

Finally, let $\order H = + \infty$.

Recall that $G$, $H$ and $K$ have the same identity element $e$ by Identity of Subgroup.

By :
 * $H = H \set e \subseteq H K$

In particular, $\order {H K} = + \infty$,

Hence:
 * $\order {H K} = \dfrac {\order H \order K} {\order {H \cap K} } = + \infty$