View source for Center (group theory)

{{Short description|Set of elements that commute with every element of a group}}
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|+ style="text-align: left;" | [[Cayley table]] for [[Dihedral group of order 8|D<sub>4</sub>]] showing elements of the center, {e, a<sup>2</sup>}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are [[Transpose|transposes]] of each other).


|-
! ∘ || e|| b||  a|| a<sup>2</sup>|| a<sup>3</sup>|| ab|| a<sup>2</sup>b|| a<sup>3</sup>b
|- align="center"
! e
| style="background: green; color: white;" | '''e'''|| b|| a|| style="background: red; color: white;" | a<sup>2</sup>|| a<sup>3</sup>|| ab|| a<sup>2</sup>b|| a<sup>3</sup>b
|- align="center"
! b
| b|| style="background: green; color: white;" | '''e'''|| a<sup>3</sup>b|| a<sup>2</sup>b|| ab|| a<sup>3</sup>|| style="background: red; color: white;" | a<sup>2</sup>|| a
|- align="center"
! a
| a|| ab|| style="background: red; color: white;" | a<sup>2</sup>|| a<sup>3</sup>|| style="background: green; color: white;" | '''e'''|| a<sup>2</sup>b|| a<sup>3</sup>b|| b
|- align="center"
! a<sup>2</sup>
| style="background: red; color: white;" | a<sup>2</sup>|| a<sup>2</sup>b|| a<sup>3</sup>|| style="background: green; color: white;" | '''e'''|| a|| a<sup>3</sup>b|| b|| ab
|- align="center"
! a<sup>3</sup>
| a<sup>3</sup>
|| a<sup>3</sup>b|| style="background: green; color: white;" | '''e'''|| a|| style="background: red; color: white;" | a<sup>2</sup>|| b|| ab|| a<sup>2</sup>b
|- align="center"
! ab
| ab|| a|| b|| a<sup>3</sup>b|| a<sup>2</sup>b|| style="background: green; color: white;" | '''e'''|| a<sup>3</sup>|| style="background: red; color: white;" | a<sup>2</sup>
|- align="center"
! a<sup>2</sup>b
| a<sup>2</sup>b|| style="background: red; color: white;" | a<sup>2</sup>|| ab|| b|| a<sup>3</sup>b|| a|| style="background: green; color: white;" | '''e'''|| a<sup>3</sup>
|- align="center"
! a<sup>3</sup>b
| a<sup>3</sup>b|| a<sup>3</sup>|| a<sup>2</sup>b|| ab|| b|| style="background: red; color: white;" | a<sup>2</sup>|| a|| style="background: green; color: white;" | '''e'''
|}
In [[abstract algebra]], the '''center''' of a [[group (mathematics)|group]] {{math|''G''}} is the [[set (mathematics)|set]] of elements that [[commutative|commute]] with every element of {{math|''G''}}. It is denoted {{math|Z(''G'')}}, from German ''[[wikt:Zentrum|Zentrum]],'' meaning ''center''. In [[set-builder notation]],

:{{math|1=Z(''G'') = {{mset|''z'' ∈ ''G'' | ∀''g'' ∈ ''G'', ''zg'' {{=}} ''gz''}}}}.

The center is a [[normal subgroup]], {{math|Z(''G'') ⊲ ''G''}}, and also a [[characteristic subgroup|characteristic]] subgroup, but is not necessarily [[fully characteristic subgroup|fully characteristic]]. The [[quotient group]], {{math|''G'' / Z(''G'')}}, is [[group isomorphism|isomorphic]] to the [[inner automorphism]] group, {{math|Inn(''G'')}}.

A group {{math|''G''}} is abelian if and only if {{math|1=Z(''G'') = ''G''}}. At the other extreme, a group is said to be '''centerless''' if {{math|Z(''G'')}} is [[trivial group|trivial]]; i.e., consists only of the [[identity element]].

The elements of the center are '''central elements'''.

==As a subgroup==
The center of ''G'' is always a [[subgroup (mathematics)|subgroup]] of {{math|''G''}}.  In particular:
# {{math|Z(''G'')}} contains the [[identity element]] of {{math|''G''}}, because it commutes with every element of {{math|''g''}}, by definition: {{math|1=''eg'' = ''g'' = ''ge''}}, where {{math|''e''}} is the identity;
# If {{math|''x''}} and {{math|''y''}} are in {{math|Z(''G'')}}, then so is {{math|''xy''}}, by associativity: {{math|1=(''xy'')''g'' = ''x''(''yg'') = ''x''(''gy'') = (''xg'')''y'' = (''gx'')''y'' = ''g''(''xy'')}} for each {{math|''g'' ∈ ''G''}}; i.e., {{math|Z(''G'')}} is closed;
# If {{math|''x''}} is in {{math|Z(''G'')}}, then so is {{math|''x''{{sup|−1}}}} as, for all {{math|''g''}} in {{math|''G''}}, {{math|''x''{{sup|−1}}}} commutes with {{math|''g''}}: {{math|1=(''gx'' = ''xg'') ⇒ (''x''{{sup|−1}}''gxx''{{sup|−1}} = ''x''{{sup|−1}}''xgx''{{sup|−1}}) ⇒ (''x''{{sup|−1}}''g'' = ''gx''{{sup|−1}})}}.

Furthermore, the center of {{math|''G''}} is always an [[abelian group|abelian]] and [[normal subgroup]] of {{math|''G''}}. Since all elements of {{math|Z(''G'')}} commute, it is closed under [[conjugate closure|conjugation]].

A group homomorphism {{math|''f'' : ''G'' → ''H''}} might not restrict to a homomorphism between their centers. The image elements  {{math|''f'' (''g'')}} commute with the image {{math|''f'' ( ''G'' )}}, but they need not commute with all of {{math|''H''}} unless {{math|''f''}}  is surjective. Thus the center mapping <math>G\to Z(G)</math> is not a functor between categories Grp and Ab, since it does not induce a map of arrows.

==Conjugacy classes and centralizers==
By definition, an element is central whenever its [[conjugacy class]] contains only the element itself; i.e. {{math|1=Cl(''g'') = {''g''}<nowiki/>}}.

The center is the [[intersection (set theory)|intersection]] of all the [[centralizer and normalizer|centralizers]] of elements of {{math|''G''}}:  <blockquote><math>Z(G) = \bigcap_{g\in G} Z_G(g).</math>   </blockquote>As centralizers are subgroups, this again shows that the center is a subgroup.

== Conjugation ==
Consider the map {{math|''f'' : ''G'' → Aut(''G'')}}, from {{math|''G''}} to the [[automorphism group]] of {{math|''G''}} defined by {{math|1=''f''(''g'') = ''ϕ''{{sub|''g''}}}}, where {{math|''ϕ''{{sub|''g''}}}} is the automorphism of {{math|''G''}} defined by 
:{{math|1=''f''(''g'')(''h'') = ''ϕ''{{sub|''g''}}(''h'') = ''ghg''{{sup|−1}}}}.

The function, {{math|''f''}} is a [[group homomorphism]], and its [[kernel (algebra)|kernel]] is precisely the center of {{math|''G''}}, and its image is called the [[inner automorphism group]] of {{math|''G''}}, denoted {{math|Inn(''G'')}}. By the [[first isomorphism theorem]] we get,
:{{math|''G''/Z(''G'') ≃ Inn(''G'')}}.

The [[cokernel]] of this map is the group {{math|Out(''G'')}} of [[outer automorphism]]s, and these form the [[exact sequence]]
:{{math|1 ⟶ Z(''G'') ⟶ ''G'' ⟶ Aut(''G'') ⟶ Out(''G'') ⟶ 1}}.

==Examples==

* The center of an [[abelian group]], {{math|''G''}}, is all of {{math|''G''}}.
* The center of the [[Heisenberg group]], {{math|''H''}}, is the set of matrices of the form: <math display="block"> \begin{pmatrix}
   1 & 0 & z\\
   0 & 1 & 0\\
   0 & 0 & 1
 \end{pmatrix}</math>
* The center of a [[nonabelian group|nonabelian]] [[simple group]] is trivial.
* The center of the [[dihedral group]], {{math|D{{sub|''n''}}}}, is trivial for odd {{math|''n'' ≥ 3}}.  For even {{math|''n'' ≥ 4}}, the center consists of the identity element together with the 180° rotation of the [[polygon]].
* The center of the [[quaternion group]], {{math|1=Q{{sub|8}} = {1, −1, i, −i, j, −j, k, −k} }}, is {{math|{1, −1}<nowiki/>}}.
* The center of the [[symmetric group]], {{math|''S''{{sub|''n''}}}}, is trivial for {{math|''n'' ≥ 3}}.
* The center of the [[alternating group]], {{math|''A''{{sub|''n''}}}}, is trivial for {{math|''n'' ≥ 4}}.
* The center of the [[general linear group]] over a [[Field (mathematics)|field]] {{math|F}}, {{math|GL{{sub|''n''}}(F)}}, is the collection of [[diagonal matrix|scalar matrices]], {{math|{{mset| sI<sub>''n''</sub> ∣ s ∈ F \ {0} }}}}.
* The center of the [[orthogonal group]], {{math|O<sub>''n''</sub>(F)}} is {{math|{I<sub>''n''</sub>, −I<sub>''n''</sub>}<nowiki/>}}.
* The center of the [[special orthogonal group]], {{math|SO(''n'')}} is the whole group when {{math|1=''n'' = 2}}, and otherwise {{math|{{mset|I<sub>''n''</sub>, −I<sub>''n''</sub>}}}} when ''n'' is even, and trivial when ''n'' is odd.
* The center of the [[unitary group]], <math>U(n)</math> is <math>\left\{ e^{i\theta} \cdot I_n \mid \theta \in [0, 2\pi) \right\}</math>.
* The center of the [[special unitary group]], <math>\operatorname{SU}(n)</math> is <math display="inline">\left\lbrace e^{i\theta} \cdot I_n \mid \theta = \frac{2k\pi}{n}, k = 0, 1, \dots, n-1 \right\rbrace </math>.
* The center of the multiplicative group of non-zero [[quaternion]]s is the multiplicative group of non-zero [[real number]]s.
* Using the [[class equation]], one can prove that the center of any non-trivial [[finite group|finite]] [[p-group]] is non-trivial.
* If the [[quotient group]] {{math|''G''/Z(''G'')}} is [[cyclic group|cyclic]], {{math|''G''}} is [[abelian group|abelian]] (and hence {{math|1=''G'' = Z(''G'')}}, so {{math|''G''/Z(''G'')}} is trivial).
* The center of the [[Rubik's Cube group]] consists of two elements – the identity (i.e. the solved state) and the [[superflip]]. The center of the [[Pocket Cube]] group is trivial.
* The center of the [[Megaminx]] group has order 2, and the center of the [[Kilominx]] group is trivial.

==Higher centers==
Quotienting out by the center of a group yields a sequence of groups called the '''[[upper central series]]''':

:{{math|1=(''G''{{sub|0}} = ''G'') ⟶ (''G''{{sub|1}} = ''G''{{sub|0}}/Z(''G''{{sub|0}})) ⟶ (''G''{{sub|2}} = ''G''{{sub|1}}/Z(''G''{{sub|1}})) ⟶ ⋯}}

The kernel of the map {{math|''G'' → ''G{{sub|i}}''}} is the '''{{math|''i''}}th center'''<ref>{{Cite journal |last=Ellis |first=Graham |date=1998-02-01 |title=On groups with a finite nilpotent upper central quotient |url=https://doi.org/10.1007/s000130050169 |journal=Archiv der Mathematik |language=en |volume=70 |issue=2 |pages=89–96 |doi=10.1007/s000130050169 |issn=1420-8938}}</ref> of {{math|''G''}} ('''second center''', '''third center''', etc.), denoted {{math|Z{{sup|''i''}}(''G'')}}.<ref>{{Cite journal |last=Ellis |first=Graham |date=1998-02-01 |title=On groups with a finite nilpotent upper central quotient |url=https://doi.org/10.1007/s000130050169 |journal=Archiv der Mathematik |language=en |volume=70 |issue=2 |pages=89–96 |doi=10.1007/s000130050169 |issn=1420-8938}}</ref> Concretely, the ({{math|''i''+1}})-st center comprises the elements that commute with all elements up to an element of the {{math|''i''}}th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to [[transfinite ordinals]] by [[transfinite induction]]; the union of all the higher centers is called the '''[[hypercenter]]'''.<ref group="note">This union will include transfinite terms if the UCS does not stabilize at a finite stage.</ref>

The [[total order#Chains|ascending chain]] of subgroups
:{{math|1 ≤ Z(''G'') ≤ Z{{sup|2}}(''G'') ≤ ⋯}}
stabilizes at ''i'' (equivalently, {{math|1=Z{{sup|''i''}}(''G'') = Z{{sup|i+1}}(''G'')}}) [[if and only if]] {{math|''G''{{sub|''i''}}}} is centerless.

===Examples===
* For a centerless group, all higher centers are zero, which is the case {{math|1=Z{{sup|0}}(''G'') = Z{{sup|1}}(''G'')}} of stabilization.
* By [[Grün's lemma]], the quotient of a [[perfect group]] by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at {{math|1=Z{{sup|1}}(''G'') = Z{{sup|2}}(''G'')}}.

==See also==
*[[Center (algebra)]]
*[[Center (ring theory)]]
*[[Centralizer and normalizer]]
*[[Conjugacy class]]

==Notes==
{{reflist|group=note}}

== References ==
* {{cite book
 | last1=Fraleigh | first1=John B. | authorlink1=
 | year = 2014
 | title = A First Course in Abstract Algebra
 | edition = 7
 | publisher = Pearson
 | isbn = 978-1-292-02496-7
}}

==External links==
* {{springer|title=Centre of a group|id=p/c021250}}

[[Category:Group theory]]
[[Category:Functional subgroups]]