I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web. This is my proof. . Since $H$ is $X−s−$permutable in $G$, then for $P$ Sylow of $G$ there exists $x \in X$ such that $P^{x}H=HP^{x}$. The Sylow of $N$ are of the form $P∩N$. Thus,$(P∩N)^{x}H=H(P∩N)^{x}$. Hence, $H$ is $X−s−$permutable in $N$.

The problem is, according to the definition in the second page, that $X \subseteq G$ but in my proof $X$ may not be a subset of $N$.