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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]
A subgroup of a group is termed a cocentral subgroup if it satisfies the following equivalent conditions:
- Its product with the center is the whole group.
- The inclusion map from the subgroup to the group induces maps between the corresponding inner automorphism groups and derived subgroup that give rise to an isoclinism.
Definition with symbols
A subgroup of a group is termed cocentral if where denotes the center of .
Relation with other properties
|property||quick description||proof of implication||proof of strictness (reverse implication failure)||intermediate notions|
|Central factor||product with centralizer equals whole group||cocentral implies central factor||central factor not implies cocentral||Right-quotient-transitively central factor|FULL LIST, MORE INFO|
|Weakly cocentral subgroup||product with center contains its normalizer|||FULL LIST, MORE INFO|
|Abelian-quotient subgroup||contains the commutator subgroup; equivalently, normal subgroup with abelian quotient group||cocentral implies abelian-quotient||abelian-quotient not implies cocentral|||FULL LIST, MORE INFO|
|Centralizer-dense subgroup||its centralizer equals its center||cocentral implies centralizer-dense||centralizer-dense not implies cocentral|||FULL LIST, MORE INFO|
|Subgroup isoclinic to the whole group||there exists an isoclinism between the subgroup and the whole group|||FULL LIST, MORE INFO|
For a group with nontrivial center, any maximal subgroup of a group is either cocentral or contains the center of the group. This idea is of use in proving that every maximal subgroup of a nilpotent group is normal. It is also related to the fact that upper join-closure of cocentrality is central factor.
|Metaproperty name||Satisfied?||Proof||Section in this article|
|Transitive subgroup property||Yes||Cocentrality is transitive||#Transitivity|
|Identity-true subgroup property||Yes||Every group is cocentral in itself||#Trimness|
|Trivially true subgroup property||No||Trivial subgroup is cocentral iff group is abelian||#Trimness|
|Intermediate subgroup condition||Yes||Cocentrality satisfies intermediate subgroup condition||#Intermediate subgroup condition|
|Upward-closed subgroup property||Yes||Cocentrality is upward-closed||#Upward-closedness|
|Finite-intersection-closed subgroup property||No||Cocentrality is not finite-intersection-closed||#Intersection-closedness|
|Upper join-closed subgroup property||No||Cocentrality is not upper join-closed||#Upper join-closedness|
The property of being cocentral is an identity-true subgroup property, that is, every group is cocentral as a subgroup of itself. However, it is not in general trivially true. In fact, the trivial subgroup is cocentral if and only if the group is abelian.
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 is cocentral in , is also cocentral in any intermediate subgroup . This follows from the fact that contains . For full proof, refer: Cocentrality satisfies intermediate subgroup condition
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
If , with a cocentral subgroup of and a cocentral subgroup of , then is cocentral in . For full proof, refer: Cocentrality is transitive
This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties
If with a cocentral subgroup of , is also a cocentral subgroup of .
For full proof, refer: cocentrality is upward-closed
This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed
It is possible to have a group and two cocentral subgroups and of such that is not a cocentral subgroup of .
For full proof, refer: Cocentrality is not finite-intersection-closed
NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.
It is possible to have a subgroup of a group that is a concentral subgroup in each of two intermediate subgroups and but not in the join .
For full proof, refer: Cocentrality is not upper join-closed
Further information: Upper join-closure of cocentrality is central factor