Central subgroup

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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]
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