Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$.
If $G$ contains an element of degree and order $p$. $G$ contains the cycle $(1,2 ... p)=g$. Let $\Delta= \lbrace p+1,p+2,...,n\rbrace$ and $\Gamma:=\lbrace1,2,...,p\rbrace$ that $\Delta ,\Gamma\subseteq \Omega$ that $\Delta\cup\Gamma=\Omega$ and $\Delta\cap\Gamma=\varnothing$.
Then can we say $\langle g\rangle$ is $p$-Sylow subgroup of $G_\Delta$ ?