In the BLS signature the subgroup $G$ of elliptic curve constructed with point $P$ with prime order $q$ by $G=\langle P\rangle $. The $h(x)$ is a hash function. The point $S$ is map (image) of $h(m)$ on elliptic curve. The signature is multiplying the private key in the point $S$.
Is the point $S$, also in $G$ and a multiple of point $P$ such as $kP$?