# Center

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

### Symbol-free definition

An element of a group is termed central if the following equivalent conditions hold:

1. It commutes with every element of the group
2. Its centralizer is the whole group
3. It is the only element in its conjugacy class. In other words, under the action of the group on itself by conjugation, it is a fixed point.
4. Under the action of the group on itself by conjugation, it fixes everything. In other words, it is in the kernel of the action of the group on itself by conjugation.

The center of a group is the set of its central elements. The center is clearly a subgroup.

Alternatively, the center of a group is defined as the kernel of the homomorphism from the group to its automorphism group, that sends each element to the corresponding inner automorphism. (see group acts as automorphisms by conjugation).

### Definition with symbols

Given a group $G$, the center of $G$, denoted $Z(G)$, is defined as the set of elements $g$ that satisfy the following equivalent conditions:

1. $gx = xg$ for all $x$ in $G$
2. $C_G(g) = G$
3. The conjugacy class of $g$ in $G$ is the singleton $\{ g \}$. In other words, under the action of $G$ on itself by conjugation, the orbit of $g$ is a one-point set -- $g$ is a fixed point.
4. For the action of $G$ on itself by conjugation, $g$ acts trivially on everything. In other words, conjugation by $g$ fixes every element.

Alternatively, $Z(G)$ is defined as the kernel of the map $G \to \operatorname{Aut}(G)$ given by $g \mapsto c_g$, where $c_g = x \mapsto gxg^{-1}$ is conjugation by $g$. (see group acts as automorphisms by conjugation).

## Group properties

The center of any group must be an abelian group. Conversely every abelian group occurs as the center of some group (in fact, of itself).

## Associated constructions

Construction Name Description
associated quotient-defining function inner automorphism group The quotient of a group by its center is isomorphic to the group of inner automorphisms, i.e. the subgroup of the automorphism group comprising those automorphisms that can be expressed using conjugation maps. This is because the map from a group to its automorphism group that sends $g$ to $c_g: x \mapsto gxg^{-1}$ is a homomorphism, and its kernel is precisely the center $Z(G)$.
associated ascending series upper central series Start with a group $G$. The upper central series of $G$ is defined as follows. Consider $Z^1(G) = Z(G)$. Let $Z^i(G)$, in general, be the inverse image in $G$ of $Z(G/Z^{i-1}(G))$ under the canonical projection $G \to G/Z^{i-1}(G)$. Essentially we are iterating the quotient-defining function that sends a group to the inner automorphism group, and taking the kernel at each step. However, we are pulling back that kernel all the way to $G$.
By convention (and commonsense) $Z^0(G)$ is the trivial group.
A group for which the upper central series terminates in finite length at the whole group is termed a nilpotent group.

## Examples

Below are some examples where the center is a proper and nontrivial subgroup. In other words, these examples exclude abelian groups (where the center is the whole group) and centerless groups (where the center is trivial):

The quotient part in the table below refers to the quotient by the center, which is isomorphic to the inner automorphism group.

Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of binary octahedral groupBinary octahedral groupCyclic group:Z2Symmetric group:S4
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of general linear group:GL(2,3)General linear group:GL(2,3)Cyclic group:Z2Symmetric group:S4
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5

## Subgroup properties

### Properties satisfied

Property Meaning Proof of satisfaction
central factor product with centralizer is whole group
central subgroup contained in the center
subgroup containing the center contains the center
normal subgroup invariant under all inner automorphisms center is normal
hereditarily normal subgroup every subgroup is normal in the whole group center is hereditarily normal
characteristic subgroup invariant under all automorphisms center is characteristic
quasiautomorphism-invariant subgroup invariant under all quasiautomorphisms center is quasiautomorphism-invariant
strictly characteristic subgroup invariant under all surjective endomorphisms center is strictly characteristic
bound-word subgroup described by a system of equations center is bound-word
purely definable subgroup can be defined in the first-order theory of the group center is purely definable
elementarily characteristic subgroup no other subgroup that is elementarily equivalently embedded center is elementarily characteristic
marginal subgroup there is a set of words such that the center is precisely the set of elements by which left or right multiplication on any letter of the word does not affect the value (the set of words here is the singleton set comprising the commutator word) center is marginal
marginal subgroup of finite type same definition as for marginal subgroup, but we insist that the set of words be finite (satisfied by the center because we can use a single word). center is marginal of finite type
finite direct power-closed characteristic subgroup in any finite direct power of the whole group, the corresponding direct power of the center is a characteristic subgroup center is finite direct power-closed characteristic
direct projection-invariant subgroup invariant under projections to direct factors center is direct projection-invariant
c-closed subgroup centralizer of some subgroup center is the centralizer of the whole group
fixed-point subgroup of a subgroup of the automorphism group fixed-point subgroup of some subgroup of the automorphism group via c-closed; explicitly, it is the fixed-point subgroup of the inner automorphism group
local powering-invariant subgroup unique $n^{th}$ root (across the whole group) of an element in the subgroup must be in the subgroup center is local powering-invariant, also via fixed-point subgroup of a subgroup of the automorphism group
powering-invariant subgroup powered for all primes that power the whole group (via local powering-invariant)
quotient-powering-invariant subgroup the quotient group is powered over all primes that the whole group is powered over. center is quotient-powering-invariant

### Properties not satisfied

The properties below are not always satisfied by the center of a group. They may be satisfied by the center for a large number of groups.

Property Meaning Proof of dissatisfaction
fully invariant subgroup invariant under all endomorphisms center not is fully invariant
1-automorphism-invariant subgroup invariant under all 1-automorphisms center not is 1-automorphism-invariant
image-closed characteristic subgroup image under any surjective homomorphism is characteristic in image of whole group center not is image-closed characteristic
intermediately characteristic subgroup characteristic in every intermediate subgroup center not is intermediately characteristic
injective endomorphism-invariant subgroup invariant under all injective endomorphisms center not is injective endomorphism-invariant
surjective endomorphism-balanced subgroup every surjective endomorphism of whole group resticts to a surjective endomorphism of subgroup center not is surjective endomorphism-balanced
weakly normal-homomorph-containing subgroup contains any normal subgroup of the whole group that is a homomorphic image of it center not is weakly normal-homomorph-containing
normality-preserving endomorphism-invariant subgroup invariant under any normality-preserving endomorphism center not is normality-preserving endomorphism-invariant
divisibility-closed subgroup if $n$ is a positive integer such that every element of the group has a $n^{th}$ root, then every element of the subgroup also has a $n^{th}$ root in the subgroup. center not is divisibility-closed (however, upper central series members are completely divisibility-closed in nilpotent group)
endomorphism kernel kernel of an endomorphism of the group. center not is endomorphism kernel
quotient-local powering-invariant subgroup If an element of the group has a unique $n^{th}$ root for some $n$, then its image modulo the subgroup in question should also have a unique $n^{th}$ root. center not is quotient-local powering-invariant (however, upper central series members are quotient-local powering-invariant in nilpotent group)
intermediately powering-invariant subgroup powering-invariant subgroup inside every intermediate subgroup center not is intermediately powering-invariant (however, upper central series members are intermediately powering-invariant in nilpotent group)
intermediately local powering-invariant subgroup local powering-invariant subgroup inside every intermediate subgroup center not is intermediately local powering-invariant (however, upper central series members are intermediately local powering-invariant in nilpotent group)

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

• Absolute center: This is the set of elements of the group fixed by every automorphism (not just by every inner automorphism).
• Epicenter: Intersection of inverse images of centers for all central extensions.
• For a group of prime power order, the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the socle of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted $\Omega_1(Z(G))$ where $G$ is the whole group, is important in many contexts. Further information: socle equals Omega-1 of center in nilpotent p-group

### Larger subgroup-defining functions

Subgroup-defining function Meaning Proof of containment Proof of strictness
second center inverse image in whole group of center of quotient by center; elements whose induced inner automorphisms commute with all inner automorphisms
Baer norm intersection of normalizers of all subgroups Baer norm contains center center not contains Baer norm
Wielandt subgroup intersection of normalizers of all subnormal subgroups (via Baer norm) (via Baer norm)

## Related subgroup properties

Property Definition in terms of center
Central subgroup contained in the center
Cocentral subgroup product with the center is the whole group
Subgroup containing the center contains the center

## Effect of operators

Operator Meaning of application to center Value
fixed-point operator a group that equals its own center abelian group
free operator a group whose center is trivial centerless group

## Subgroup-defining function properties

Property name Satisfied? Proof Statement with symbols
reverse monotone subgroup-defining function Yes Suppose $H \le G$. Then, $H \cap Z(G) \le Z(H)$.
idempotent subgroup-defining function Yes For any group $G$, $Z(Z(G)) = Z(G)$, i.e., the center of the center is the center.

## In groups with additional structure

### Topological group

The center of a T0 topological group is always a closed subgroup. Thus, any topologically simple group must be either centerless or abelian.

For full proof, refer: center is closed subgroup

### Algebra group

The center of an algebra group must be an algebra subgroup.

For full proof, refer: center of algebra group is algebra subgroup

## Computation

### GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Center
View other GAP-computable subgroup-defining functions

To compute the center of a group in GAP, the syntax is:

Center (group);

where group could either be an on-the-spot description of the group or a name alluding to a previously defined group.

We can assign this as a value, to a new name, for instance:

zg := Center (g);

where g is the original group and zg is the center.