Weakly closed subgroup of Sylow subgroup

As a property of a subgroup in a Sylow subgroup
Suppose $$G$$ is a finite group, $$P$$ is a $$p$$-Sylow subgroup of $$G$$ and $$H$$ is a subgroup of $$P$$. The triple $$H \le P \le G$$ denotes a weakly closed subgroup of Sylow subgroup if $$H$$ is a defining ingredient::weakly closed subgroup of $$P$$ relative to $$G$$. In other words, any conjugate of $$H$$ by an element of $$G$$, that is contained in $$P$$, is equal to $$H$$.

As a subgroup property
A subgroup of a finite group is termed a weakly closed subgroup of Sylow subgroup if it satisfies the following equivalent conditions:


 * It occurs as a defining ingredient::weakly closed subgroup of a defining ingredient::Sylow subgroup.
 * It occurs as a weakly closed subgroup in every Sylow subgroup of the group containing it.

Stronger properties

 * Weaker than::Sylow subgroup

Weaker properties

 * Stronger than::p-subgroup that is normal in every p-Sylow subgroup containing it
 * Stronger than::Normal subgroup of Sylow subgroup