Editing Center (group theory) (section)

{{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 [[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).


|-
! <math>\circ</math> || 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)\triangleleft G</math>, 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'''.