Let $G$ be a group, if $x\in G$ has order $p$, for some prime $p$, and $A\space per\space G$ (that is, A is a permutable subgroup of $G$), then I want to show that $x$ normalizes $A$.
Definition (pag. 393): A subgroup H is said to be permutable in a group G if HK=KH whenever K≤G.
Exercise 6 (pag. 396): A permutable subgroup is normalized by every element of prime order.