Weakly critical subgroup

Symbol-free definition
A subgroup of a group of prime power order (or more generally, of a possibly infinite p-group) is termed a critical subgroup if it satisfies the following three conditions:


 * 1) The subgroup is a Frattini-in-center group: Its Frattini subgroup is contained in its center.
 * 2) The subgroup is a commutator-in-center subgroup: Its commutator with the whole group is contained in its center.
 * 3) The subgroup is a self-centralizing subgroup: Its centralizer in the whole group is contained in it.

Definition with symbols
Let $$G$$ be a group of prime power order (or more generally, a possibly infinite p-group).

A subgroup $$H$$ of $$G$$ is said to be critical if the following three conditions hold:


 * 1) $$\Phi(H) \le Z(H)$$, i.e., the Frattini subgroup is contained inside the center (i.e., $$H$$ is a defining ingredient::Frattini-in-center group).
 * 2) $$[G,H] \le Z(H)$$ (i.e., $$H$$ is a defining ingredient::commutator-in-center subgroup of $$G$$).
 * 3) $$C_G(H)= Z(H)$$ (i.e., $$H$$ is a self-centralizing subgroup of $$G$$).

Stronger properties

 * Weaker than::Maximal among abelian normal subgroups: . Note that the other two conditions are true for all abelian normal subgroups.
 * Weaker than::Critical subgroup
 * Weaker than::Potentially critical subgroup

Weaker properties

 * Stronger than::Commutator-in-center subgroup
 * Stronger than::Self-centralizing normal subgroup
 * Stronger than::Class two normal subgroup
 * Stronger than::Coprime automorphism-faithful normal subgroup
 * Stronger than::Coprime automorphism-faithful subgroup

Metaproperties
If $$H \le K \le G$$ such that $$H$$ is weakly critical in $$G$$, then $$H$$ is weakly critical in $$K$$.