Let $$H,K$$ be finite subgroups of a group $$G$$. Show that
$|HK|=\frac{|H|\cdot|K|}{|H\cap K|}.$

$\pi\colon (H\times K)\times HK\to HK$

$((h,k),x)\mapsto hxk^{-1}.$

$H\times (G/K)_l\to (G/K)_l$

$(h,xK)\mapsto hxK.$

$|Orb(K)|=\frac{|HK|}{|K|}.$

$\frac{|HK|}{|K|}=\frac{|H|}{|H\cap K|},$

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