# Weakly critical subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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:

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

### Definition with symbols

Let be a group of prime power order (or more generally, a possibly infinite p-group).

A subgroup of is said to be **critical** if the following three conditions hold:

- , i.e., the Frattini subgroup is contained inside the center (i.e., is a Frattini-in-center group).
- (i.e., is a commutator-in-center subgroup of ).
- (i.e., is a self-centralizing subgroup of ).

## 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If such that is weakly critical in , then is weakly critical in .