P-stable group

Definition
Let $$G$$ be a finite group and $$p$$ be a prime number. We say that $$G$$ is a $$p$$-stable group if either $$G$$ is trivial or $$G$$ has a nontrivial normal $$p$$-subgroup and $$G$$ satisfies the following:

Suppose $$P$$ is a $$p$$-subgroup of $$G$$ such that $$O_{p'}(G)P$$ is normal in $$G$$. Then, if $$A$$ is a $$p$$-subgroup of $$N_G(P)$$ with the property that $$[[P,A],A]$$ is trivial, we have:

$$AC_G(P)/C_G(P) \le O_p(N_G(P)/C_G(P))$$.

Stronger properties

 * Weaker than::Strongly p-solvable group: