# Central 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]
This article describes a property that arises as the conjunction of a subgroup property: central factor with a group property (itself viewed as a subgroup property): Abelian group
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## Definition

### Equivalent definitions in tabular format

Note that each of these definitions assumes that we already have a group and a subgroup.

No. Shorthand A subgroup of a group is normal in it if... A subgroup $H$ of a group $G$ is central in $G$ if ...
1 inside the center it is inside the center of the whole group. $H \le Z(G)$ where $Z(G)$ is the center of $G$.
2 fixed under inner automorphisms every inner automorphism of the whole group fixes every element of the subgroup. $ghg^{-1} = h$ for all $g \in G$ and $h \in H$.
3 commutes element-wise with whole group every element of the subgroup commutes with every element of the whole group. $gh = hg$ for all $g \in G$ and $h \in H$.
4 abelian central factor it is a central factor of the whole group and is also abelian. $HC_G(H) = G$, where $C_G(H)$ is the centralizer of $H$ in $G$, and also $H$ is abelian.

### Equivalence of definitions

The equivalence of definitions (1), (2) and (3) follows directly from the definition of center. For the equivalence with (4), refer abelian central factor equals central subgroup.

## Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Metaproperty name Satisfied? Proof Statement with symbols
abelian-tautological subgroup property Yes abelian group is its own center If $G$ is an abelian group and $H$ is a subgroup of $G$, then $H$ is central in $G$.
left-hereditary subgroup property Yes direct from definition If $H \le K \le G$ are groups such that $K$ is central in $G$, then $H$ is also central in $G$.
intersection-closed subgroup property Yes (via left-hereditary) If $G$ is a group and $H_i, i \in I$, is a nonempty collection of central subgroups of $G$, then the intersection $\bigcap_{i \in I} H_i$ is also a central subgroup of $G$.
strongly join-closed subgroup property Yes the join is still in the center If $G$ is a group and $H_i, i \in I$, is a (possibly empty) collection of central subgroups of $G$, then the join $\left \langle H_i \right \rangle_{i \in I}$ is also a central subgroup of $G$.
intermediate subgroup condition Yes direct from definition If $H \le K \le G$ are groups such that $H$ is a central subgroup of $G$, then $H$ is also a central subgroup of $K$.
image condition Yes direct from definition If $H$ is a central subgroup of $G$ and $\varphi:G \to K$ is a surjective homomorphism of groups, then $\varphi(H)$ is central in $K$.
identity-true subgroup property (and hence trim subgroup property) No non-abelian groups exist It is possible to have a group $G$ that is not a central subgroup of itself. In fact, we can choose any non-abelian group as an example.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup of abelian group the whole group is abelian |FULL LIST, MORE INFO
abelian direct factor the subgroup is abelian and is a direct factor of the whole group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup Abelian normal subgroup, Amalgam-characteristic subgroup, Amalgam-strictly characteristic subgroup, Center-fixing automorphism-invariant subgroup, Class two normal subgroup, Commutator-in-center subgroup, Conjugacy-closed normal subgroup, Dedekind normal subgroup, Direct factor over central subgroup, Hereditarily normal subgroup, Join-transitively central factor, Nilpotent normal subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO
central factor product with centralizer is whole group Intersection-transitively central factor, Join of direct factor and central subgroup, Join-transitively central factor|FULL LIST, MORE INFO
hereditarily normal subgroup (also called quasicentral subgroup) every subgroup of it is normal in the whole group Abelian hereditarily normal subgroup|FULL LIST, MORE INFO
SCAB-subgroup every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup Hereditarily normal subgroup, Join-transitively central factor|FULL LIST, MORE INFO
transitively normal subgroup every normal subgroup of it is normal in the whole group Abelian hereditarily normal subgroup, Conjugacy-closed normal subgroup, Hereditarily normal subgroup, Join-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
conjugacy-closed subgroup any two elements in it that are conjugate in the whole group are conjugate in the subgroup Conjugacy-closed normal subgroup, Subset-conjugacy-closed subgroup|FULL LIST, MORE INFO
conjugacy-closed normal subgroup normal and conjugacy-closed Join-transitively central factor|FULL LIST, MORE INFO
abelian normal subgroup abelian as a group and normal in the whole group |FULL LIST, MORE INFO

## Formalisms

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

A function restriction expression for the property of being a central subgroup is as follows:

Inner automorphism $\to$ Identity map

### First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

The subgroup property of being central can be expressed in first-order language as follows: $H$ is central in $G$ if and only if:

$\forall g \in G, h \in H: \ gh = hg$