Editing Center (group theory) (section)

==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.