Central subgroup

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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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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.	$H\leq Z(G)$ 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 $H\leq K\leq G$ 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 $H\leq K\leq G$ 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|FULL LIST, MORE INFO
central factor	product with centralizer is whole group			|FULL LIST, MORE INFO
hereditarily normal subgroup (also called quasicentral subgroup)	every subgroup of it is normal in the whole group			|FULL LIST, MORE INFO
SCAB-subgroup	every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup			|FULL LIST, MORE INFO
transitively normal subgroup	every normal subgroup of it is normal in the whole group			|FULL LIST, MORE INFO
conjugacy-closed subgroup	any two elements in it that are conjugate in the whole group are conjugate in the subgroup			|FULL LIST, MORE INFO
conjugacy-closed normal subgroup	normal and conjugacy-closed			|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.
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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:

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