I would like to know if this statement (which i just met and suspiciously never realized before) and its proof are true:
Let $p$, $q$ be distinct primes and $G$ a group of order $n=p^{\alpha}q^{\beta}$. If $H$ is a $p$-Sylow subgroup of $G$ and $K$ a $q$-Sylow subgroup, then $G=HK$.
$\textit{Proof : }$ the order of the set $HK$ is $\frac{|H||K|}{|H\cap K|}=p^{\alpha}q^{\beta}=|G|$.
I'm a bit surprised by that.