Center (group theory)

{{Short description|Set of elements that commute with every element of a group}} {{Use American English|date=March 2021}} {{Use mdy dates|date=March 2021}} {{redirect|Group center|the American counter-cultural group|Aldo Tambellini#Lower East Side artists}} {| class="wikitable floatright"

|+ style="text-align: left;" | Cayley table for D₄ showing elements of the center, {e, a²}, 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 transposes of each other).

|- ! <math>\circ</math> || e|| b|| a|| a²|| a³|| ab|| a²b|| a³b |- align="center" ! e | style="background: green; color: white;" | '''e'''|| b|| a|| style="background: red; color: white;" | a²|| a³|| ab|| a²b|| a³b |- align="center" ! b | b|| style="background: green; color: white;" | '''e'''|| a³b|| a²b|| ab|| a³|| style="background: red; color: white;" | a²|| a |- align="center" ! a | a|| ab|| style="background: red; color: white;" | a²|| a³|| style="background: green; color: white;" | '''e'''|| a²b|| a³b|| b |- align="center" ! a² | style="background: red; color: white;" | a²|| a²b|| a³|| style="background: green; color: white;" | '''e'''|| a|| a³b|| b|| ab |- align="center" ! a³ | a³ || a³b|| style="background: green; color: white;" | '''e'''|| a|| style="background: red; color: white;" | a²|| b|| ab|| a²b |- align="center" ! ab | ab|| a|| b|| a³b|| a²b|| style="background: green; color: white;" | '''e'''|| a³|| style="background: red; color: white;" | a² |- align="center" ! a²b | a²b|| style="background: red; color: white;" | a²|| ab|| b|| a³b|| a|| style="background: green; color: white;" | '''e'''|| a³ |- align="center" ! a³b | a³b|| a³|| a²b|| ab|| b|| style="background: red; color: white;" | a²|| a|| style="background: green; color: white;" | '''e''' |} In abstract algebra, the '''center''' of a group {{math|''G''}} is the set of elements that commute with every element of {{math|''G''}}. It is denoted {{math|Z(''G'')}}, from German ''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)\triangleleft G</math>, and also a characteristic subgroup, but is not necessarily fully characteristic. The quotient group, {{math|''G'' / Z(''G'')}}, is 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; 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 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 and normal subgroup of {{math|''G''}}. Since all elements of {{math|Z(''G'')}} commute, it is closed under 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 of all the 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 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 automorphisms, 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 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 {{math|F}}, {{math|GL{{sub|''n''}}(F)}}, is the collection of scalar matrices, {{math|{{mset| sI_''n'' ∣ s ∈ F \ {0} }}}}. * The center of the orthogonal group, {{math|O_''n''(F)}} is {{math|{I_''n'', −I_''n''}<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_''n'', −I_''n''}}}} 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 quaternions is the multiplicative group of non-zero real numbers. * Using the class equation, one can prove that the center of any non-trivial finite p-group is non-trivial. * If the quotient group {{math|''G''/Z(''G'')}} is cyclic, {{math|''G''}} is 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|url-access=subscription }}</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|url-access=subscription }}</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 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