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]

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Definition

Symbol-free definition

A subgroup of a group is termed a cocentral subgroup if it satisfies the following equivalent conditions:

1. Its product with the center is the whole group.
2. 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 $H$ of a group $G$ is termed cocentral if $HZ(G) = G$ where $Z(G)$ denotes the center of $G$.

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 $H$ is cocentral in $G$, $H$ is also cocentral in any intermediate subgroup $K$. This follows from the fact that $Z(K)$ contains $K \cap Z(G)$. 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 $H \le K \le G$, with $H$ a cocentral subgroup of $K$ and $K$ a cocentral subgroup of $G$, then $H$ is cocentral in $G$. 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 $H \le K \le G$ with $H$ a cocentral subgroup of $G$, $K$ is also a cocentral subgroup of $G$.

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 $G$ and two cocentral subgroups $H_1$ and $H_2$ of $G$ such that $H_1 \cap H_2$ is not a cocentral subgroup of $G$.

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 $H$ of a group $G$ that is a concentral subgroup in each of two intermediate subgroups $K_1$ and $K_2$ but not in the join $\langle K_1, K_2 \rangle$.

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