[[Cayley table]] for [[Dihedral group of order 8|D<sub>4</sub>]] showing elements of the center, {e, a<sup>2</sup>}, arranged symmetrically about the main diagonal (illustrating they each commute with all other elements)
In abstract algebra, the center of a group, , is the set of elements that commute with every element of .
It is denoted , from German Zentrum, meaning center.
In set-builder notation,
.
The center is a normal subgroup, .
As a subgroup, it is always characteristic, but is not necessarily fully characteristic.
The quotient group, , is isomorphic to the inner automorphism group, .
A group  is abelian if and only if .
At the other extreme, a group is said to be centerless if  is trivial; i.e., consists only of the identity element.
The elements of the center are sometimes called central.
As a subgroup
The center of G is always a subgroup of .
In particular:
contains the identity element of , because it commutes with every element of , by definition: , where  is the identity;
If  and  are in , then so is , by associativity:  for each ; i.e.,  is closed;
If  is in , then so is  as, for all  in ,  commutes with : .
Furthermore, the center of  is always a normal subgroup of .
Since all elements of  commute, it is closed under conjugation.
Conjugacy classes and centralizers
By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., .
The center is also the intersection of all the centralizers of each element of .
As centralizers are subgroups, this again shows that the center is a subgroup.
Conjugation
Consider the map, , from  to the automorphism group of  defined by , where  is the automorphism of  defined by
.
The function,  is a group homomorphism, and its kernel is precisely the center of , and its image is called the inner automorphism group of , denoted .
By the first isomorphism theorem we get,
.
The cokernel of this map is the group  of outer automorphisms, and these form the exact sequence
.
Examples
The center of an abelian group, , is all of .
The center of the Heisenberg group, , is the set of matrices of the form:  \begin{pmatrix}    1 & 0 & z\\    0 & 1 & 0\\    0 & 0 & 1  \end{pmatrix}
The center of a nonabelian simple group is trivial.
The center of the dihedral group, , is trivial for odd .
For even , the center consists of the identity element together with the 180° rotation of the polygon.
The center of the quaternion group, , is .
The center of the symmetric group, , is trivial for .
The center of the alternating group, , is trivial for .
The center of the general linear group over a field , , is the collection of scalar matrices, .
The center of the orthogonal group,  is .
The center of the special orthogonal group,  is the whole group when , and otherwise  when n is even, and trivial when n is odd.
The center of the unitary group, U(n) is \left\{ e^{i\theta} \cdot I_n \mid \theta \in [0, 2\pi) \right\}.
The center of the special unitary group, \operatorname{SU}(n) is \left\lbrace e^{i\theta} \cdot I_n \mid \theta = \frac{2k\pi}{n}, k = 0, 1, \dots, n-1 \right\rbrace .
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  is cyclic,  is abelian (and hence , so  is trivial).
The center of the megaminx group is a cyclic group of 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:
The kernel of the map  is the th center of  (second center, third center, etc.) and is denoted .
Concretely, the ()-st center are the terms that commute with all elements up to an element of the 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.This union will include transfinite terms if the UCS does not stabilize at a finite stage.
The ascending chain of subgroups
stabilizes at i (equivalently, ) if and only if  is centerless.
Examples
For a centerless group, all higher centers are zero, which is the case  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 .
See also
Center (algebra)
Center (ring theory)
Centralizer and normalizer
Conjugacy class
Notes
References
External links
