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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:

It commutes with every element of the group
Its centralizer is the whole group
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.
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:

$gx=xg$ for all $x$ in $G$
$C_{G}(g)=G$
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.
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 dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

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 $H\leq G$ . Then, $H\cap Z(G)\leq Z(H)$ .
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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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

External links

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Definition links

@@ Line 32: / Line 32: @@
 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==
+{| class="sortable" border="1"
+! 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 automorphism]]s, 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 <math>g</math> to <math>c_g: x \mapsto gxg^{-1}</math> is a homomorphism, and its kernel is precisely the center <math>Z(G)</math>.
+|-
+| associated [[ascending series]] || [[upper central series]] || Start with a group <math>G</math>. The upper central series of <math>G</math> is defined as follows. Consider <math>Z^1(G) = Z(G)</math>. Let <math>Z^i(G)</math>, in general, be the inverse image in <math>G</math> of <math>Z(G/Z^{i-1}(G))</math> under the canonical projection <math>G \to G/Z^{i-1}(G)</math>. 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 <math>G</math>. <br>By convention (and commonsense) <math>Z^0(G)</math> is the [[trivial group]].<br> 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 group]]s (where the center is the whole group) and [[centerless group]]s (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]].
+{{sdf examples}}
 ==Subgroup properties==
@@ Line 39: / Line 56: @@
 ! Property !! Meaning !! Proof of satisfaction
 |-
-| [[Satisfies property::Central factor]] || product with centralizer is whole group ||
+| [[Satisfies property::central factor]] || product with centralizer is whole group ||
+|-
+| [[Satisfies property::central subgroup]] || contained in the center ||
 |-
-| [[Satisfies property::Central subgroup]] || contained in the center ||
+| [[Satisfies property::subgroup containing the center]] || contains the center ||
 |-
-| [[Satisfies property::Normal subgroup]] || invariant under all [[inner automorphism]]s || [[center is normal]]
+| [[Satisfies property::normal subgroup]] || invariant under all [[inner automorphism]]s || [[center is normal]]
 |-
-| [[Satisfies property::Hereditarily normal subgroup]] || every subgroup is [[normal subgroup|normal]] in the whole group || [[center is hereditarily normal]]
+| [[Satisfies property::hereditarily normal subgroup]] || every subgroup is [[normal subgroup|normal]] in the whole group || [[center is hereditarily normal]]
 |-
-| [[Satisfies property::Characteristic subgroup]] || invariant under all [[automorphism]]s || [[center is characteristic]]
+| [[Satisfies property::characteristic subgroup]] || invariant under all [[automorphism]]s || [[center is characteristic]]
 |-
-| [[Satisfies property::Quasiautomorphism-invariant subgroup]] || invariant under all [[quasiautomorphism]]s || [[center is quasiautomorphism-invariant]]
+| [[Satisfies property::quasiautomorphism-invariant subgroup]] || invariant under all [[quasiautomorphism]]s || [[center is quasiautomorphism-invariant]]
 |-
-| [[Satisfies property::Strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[center is strictly characteristic]]
+| [[Satisfies property::strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[center is strictly characteristic]]
 |-
-| [[Satisfies property::Bound-word subgroup]] || described by a system of equations || [[center is bound-word]]
+| [[Satisfies property::bound-word subgroup]] || described by a system of equations || [[center is bound-word]]
+|-
+| [[Satisfies property::purely definable subgroup]] || can be defined in the first-order theory of the group || [[center is purely definable]]
+|-
+| [[Satisfies property::elementarily characteristic subgroup]]|| no other subgroup that is [[elementarily equivalently embedded subgroups|elementarily equivalently embedded]] || [[center is elementarily characteristic]]
+|-
+| [[Satisfies property::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]]
+|-
+| [[Satisfies property::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]]
+|-
+| [[Satisfies property::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]]
+|-
+| [[Satisfies property::direct projection-invariant subgroup]] || invariant under projections to direct factors || [[center is direct projection-invariant]]
+|-
+| [[satisfies property::c-closed subgroup]] || centralizer of some subgroup || center is the centralizer of the whole group
+|-
+| [[satisfies property::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]]
+|-
+| [[satisfies property::local powering-invariant subgroup]] || unique <math>n^{th}</math> root (across the whole group) of an element in the subgroup must be in the subgroup || [[center is local powering-invariant]], also [[fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant|via fixed-point subgroup of a subgroup of the automorphism group]]
+|-
+| [[satisfies property::powering-invariant subgroup]] || powered for all primes that power the whole group || (via local powering-invariant)
+|-
+| [[satisfies property::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.
 {| class="sortable" border="1"
 ! Property !! Meaning !! Proof of dissatisfaction
 |-
-| [[Dissatisfies property::Fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[center not is fully invariant]]
+| [[Dissatisfies property::fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[center not is fully invariant]]
 |-
 | [[Dissatisfies property::1-automorphism-invariant subgroup]] || invariant under all [[1-automorphism]]s || [[center not is 1-automorphism-invariant]]
 |-
-| [[Dissatisfies property::Image-closed characteristic subgroup]] || image under any surjective homomorphism is characteristic in image of whole group || [[center not is image-closed characteristic]]
+| [[Dissatisfies property::image-closed characteristic subgroup]] || image under any surjective homomorphism is characteristic in image of whole group || [[center not is image-closed characteristic]]
+|-
+| [[Dissatisfies property::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || [[center not is intermediately characteristic]]
+|-
+| [[Dissatisfies property::injective endomorphism-invariant subgroup]] || invariant under all injective endomorphisms || [[center not is injective endomorphism-invariant]]
+|-
+| [[Dissatisfies property::surjective endomorphism-balanced subgroup]] || every surjective endomorphism of whole group resticts to a surjective endomorphism of subgroup || [[center not is surjective endomorphism-balanced]]
+|-
+| [[Dissatisfies property::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]]
+|-
+| [[Dissatisfies property::normality-preserving endomorphism-invariant subgroup]] || invariant under any [[normality-preserving endomorphism]] || [[center not is normality-preserving endomorphism-invariant]]
 |-
-| [[Dissatisfies property::Intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || [[center not is intermediately characteristic]]
+| [[Dissatisfies property::divisibility-closed subgroup]] || if <math>n</math> is a positive integer such that every element of the group has a <math>n^{th}</math> root, then every element of the subgroup also has a <math>n^{th}</math> root in the subgroup. || [[center not is divisibility-closed]] (however, [[upper central series members are completely divisibility-closed in nilpotent group]])
 |-
-| [[Dissatisfies property::Injective endomorphism-invariant subgroup]] || invariant under all injective endomorphisms || [[center not is injective endomorphism-invariant]]
+| [[Dissatisfies property::endomorphism kernel]] || kernel of an endomorphism of the group. || [[center not is endomorphism kernel]]
 |-
-| [[Dissatisfies property::Surjective endomorphism-balanced subgroup]] || every surjective endomorphism of whole group resticts to a surjective endomorphism of subgroup || [[center not is surjective endomorphism-balanced]]
+| [[dissatisfies property::quotient-local powering-invariant subgroup]] || If an element of the group has a unique <math>n^{th}</math> root for some <math>n</math>, then its image modulo the subgroup in question should also have a unique <math>n^{th}</math> root. || [[center not is quotient-local powering-invariant]] (however, [[upper central series members are quotient-local powering-invariant in nilpotent group]])
+|-
+| [[dissatisfies property::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]])
+|-
+| [[dissatisfies property::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]])
 |}
@@ Line 78: / Line 135: @@
 ===Smaller subgroup-defining functions===
-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 <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts.
+* [[Contains::Absolute center]]: This is the set of elements of the group fixed by ''every'' [[automorphism]] (not just by every [[inner automorphism]]).
+* [[Contains::Epicenter]]: Intersection of inverse images of centers for all [[central extension]]s.
+* 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 <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts. {{further|[[socle equals Omega-1 of center in nilpotent p-group]]}}
 ===Larger subgroup-defining functions===
@@ Line 106: / Line 165: @@
 ==Effect of operators==
-===Fixed-point operator===
+{| class="sortable" border="1"
+! Operator !! Meaning of application to center !! Value
-A group equals its own center if and only if it is an [[Abelian group]].
+|-
+| fixed-point operator || a group that equals its own center || [[abelian group]]
-===Free operator===
+|-
+| free operator || a group whose center is trivial || [[centerless group]]
-A group whose center is trivial is termed a [[centerless group]].
+|}
 ==Subgroup-defining function properties==
-{{reverse monotone sdf}}
+{| class="sortable" border="1"
+! Property name !! Satisfied? !! Proof !! Statement with symbols
-The '''center''' [[subgroup-defining function]] is reverse monotone. That is:
+|-
+| [[satisfies property::reverse monotone subgroup-defining function]] || Yes || || Suppose <math>H \le G</math>. Then, <math>H \cap Z(G) \le Z(H)</math>.
-Let <math>H</math> &le; <math>G</math> be groups. Then, <math>Z(H)</math> contains the group <math>Z(G)</math> &cap; <math>H</math>.
+|-
+| [[satisfies property::idempotent subgroup-defining function]] || Yes || || For any group <math>G</math>, <math>Z(Z(G)) = Z(G)</math>, i.e., the center of the center is the center.
-{{idempotent sdf}}
+|}
-The center of the center is the center. This is because the center is an [[Abelian group]], and the center of any Abelian group is itself.
 ==In groups with additional structure==
@@ Line 130: / Line 187: @@
 ===Topological group===
-The center of a [[T0 topological group]] is always a [[closed subgroup]]. Thus, any [[topologically simple group]] must be either [[centerless group|centerless]] or [[Abelian group|Abelian]].
+The center of a [[T0 topological group]] is always a [[closed subgroup]]. Thus, any [[topologically simple group]] must be either [[centerless group|centerless]] or [[abelian group|abelian]].
 {{proofat|[[center is closed subgroup]]}}
-==Associated constructions==
+===Algebra group===
-{{associated qdf|Inner automorphism group}}
-The quotient of a group by its center is isomorphic to the group of [[inner automorphism]]s, This is because the map fro ma group to its automorphism group that sends <math>g</math> to <math>c_g: x \mapsto gxg^{-1}</math> is a homomorphism, and its kernel is precisely the center <math>Z(G)</math>.
-{{associated as|Upper central series}}
-Start with a group <math>G</math>. Consider <math>Z^1(G) = Z(G)</math>. Let <math>Z^i(G)</math>, in general, be the inverse image in <math>G</math> of <math>Z(G/Z^{i-1}(G))</math> under the canonical projection <math>G \to G/Z^{i-1}(G)</math>. 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 <math>G</math>.
+The center of an [[algebra group]] must be an [[algebra subgroup]].
-By convention (and commonsense) <math>Z^0(G)</math> is the [[trivial group]].
+{{proofat|[[center of algebra group is algebra subgroup]]}}
-A group for which the upper central series terminates in finite length at the whole group, is termed a [[nilpotent group]].
 ==Computation==
@@ Line 157: / Line 207: @@
 To compute the center of a group in GAP, the syntax is:
-<pre>Center (group);</pre>
+<tt>Center (group);</tt>
-where <pre>group</pre> could either be an on-the-spot description of the group or a name aluding to a previously defined group.
+where <tt>group</tt> 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:
-<pre>zg = DerivedSubgroup (g);</pre>
+<tt>zg := Center (g);</tt>
+where <tt>g</tt> is the original group and <tt>zg</tt> is the center.
-where <pre>g</pre> is the original group and <pre>zg</pre> is the center.
 ==References==
 ===Textbook references===