Weakly critical subgroup
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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]
- 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
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
- 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
- Commutator-in-center subgroup
- Self-centralizing normal subgroup
- Class two normal subgroup
- Coprime automorphism-faithful normal subgroup
- Coprime automorphism-faithful subgroup
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
ABOUT 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 .