Cocentral 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]
Definition
Symbol-free definition
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
Weaker 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 |
Dichotomy
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.
Metaproperties
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 |
Trimness
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
Transitivity
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
Upward-closedness
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
Intersection-closedness
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
Upper join-closedness
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
Effect of property operators
The upper join-closure operator
Applying the upper join-closure operator to this property gives: central factor