Central subgroup
From Groupprops
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
View a complete list of such conjunctions
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 of a group is central in if ... |
---|---|---|---|
1 | inside the center | it is inside the center of the whole group. | where is the center of . |
2 | fixed under inner automorphisms | every inner automorphism of the whole group fixes every element of the subgroup. | for all and . |
3 | commutes element-wise with whole group | every element of the subgroup commutes with every element of the whole group. | for all and . |
4 | abelian central factor | it is a central factor of the whole group and is also abelian. | , where is the centralizer of in , and also 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 is an abelian group and is a subgroup of , then is central in . |
left-hereditary subgroup property | Yes | direct from definition | If are groups such that is central in , then is also central in . |
intersection-closed subgroup property | Yes | (via left-hereditary) | If is a group and , is a nonempty collection of central subgroups of , then the intersection is also a central subgroup of . |
strongly join-closed subgroup property | Yes | the join is still in the center | If is a group and , is a (possibly empty) collection of central subgroups of , then the join is also a central subgroup of . |
intermediate subgroup condition | Yes | direct from definition | If are groups such that is a central subgroup of , then is also a central subgroup of . |
image condition | Yes | direct from definition | If is a central subgroup of and is a surjective homomorphism of groups, then is central in . |
identity-true subgroup property (and hence trim subgroup property) | No | non-abelian groups exist | It is possible to have a group 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
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 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: is central in if and only if: