Parabolic subgroup for a prime

Definition with symbols
Given a finite group $$G$$ and a prime $$p$$, a subgroup $$H$$ of $$G$$ is said to be $$p$$-parabolic if it satisfies the following condition: let $$P = O_p(H)$$, that is, the Sylow core (largest normal $$p$$-subgroup) of $$H$$. Then, $$H = N_G(P)$$.

Stronger properties

 * Sylow normalizer