R-normal subgroup

Symbol-free definition
A subgroup of a finite group is termed R-normal if it commutes with every element whose order is relatively prime to the order of the subgroup.

Definition with symbols
A subgroup $$H$$ of a finite group $$G$$ is termed R-normal if for any $$x$$ such that the orders of $$x$$ and $$H$$ are relatively prime, $$Hx = xH$$.

Stronger properties

 * Weaker than::Normal subgroup

Intersection-closedness
Is it true that an intersection of R-normal subgroups is R-normal?