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 part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of binary octahedral group	Binary octahedral group	Cyclic group:Z2	Symmetric group:S4
Center of central product of D8 and Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of general linear group:GL(2,3)	General linear group:GL(2,3)	Cyclic group:Z2	Symmetric group:S4
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating 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

The computation problem

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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.

References

Textbook references

• Topics in Algebra by I. N. Herstein, ^{More info}, Page 47
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 14 (definition introduced in paragraph)
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 50
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 52, Point (4.10)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 26, Automorphisms
• Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 5 (definition in paragraph, as a special case of centralizer)
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 14 (definition in paragraph)
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 34 (definition introduced in Exercise 11)
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 75, Exercise 52(b) (definition introduced in exercise, as a special case of centralizer, defined implicitly)
• Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 61

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