Weakly critical subgroup

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Definition

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 Frattini-in-center group).
2. $[G,H] \le Z(H)$ (i.e., $H$ is a commutator-in-center subgroup of $G$).
3. $C_G(H)= Z(H)$ (i.e., $H$ is a self-centralizing subgroup of $G$).

Relation with other properties

Stronger properties

• Maximal among abelian normal subgroups: For full proof, refer: Maximal among abelian normal implies self-centralizing in nilpotent. Note that the other two conditions are true for all abelian normal subgroups.
• Critical subgroup
• Potentially critical subgroup

Weaker properties

• Commutator-in-center subgroup
• Self-centralizing normal subgroup
• Class two normal subgroup
• Coprime automorphism-faithful normal subgroup
• Coprime automorphism-faithful subgroup

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
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If $H \le K \le G$ such that $H$ is weakly critical in $G$, then $H$ is weakly critical in $K$.