Editing Center (group theory) (section)

==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 [[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'')}}.