# Center (group theory)

In abstract algebra, the **center** of a group *G*, denoted *Z*(*G*),^{[note 1]} is the set of elements that commute with every element of *G*. In set-builder notation,

The center is a subgroup of *G*, which by definition is abelian (that is, commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group *G* / *Z*(*G*) is isomorphic to the group of inner automorphisms of *G*.

A group *G* is abelian if and only if *Z*(*G*) = *G*. At the other extreme, a group is said to be **centerless** if *Z*(*G*) is trivial, i.e. consists only of the identity element.

The elements of the center are sometimes called **central**.

## Contents

## As a subgroup

The center of *G* is always a subgroup of *G*. In particular:

*Z*(*G*) contains*e*, the identity element of*G*, because*eg*=*g*=*ge*for all*g*∈ G by definition of*e*, so by definition of*Z*(*G*),*e*∈*Z*(*G*);- If
*x*and*y*are in*Z*(*G*), then (*xy*)*g*=*x*(*yg*) =*x*(*gy*) = (*xg*)*y*= (*gx*)*y*=*g*(*xy*) for each*g*∈*G*, and so*xy*is in*Z*(*G*) as well (i.e.,*Z*(*G*) exhibits closure); - If
*x*is in*Z*(*G*), then*gx*=*xg*, and multiplying twice, once on the left and once on the right, by*x*^{−1}, gives*x*^{−1}*g*=*gx*^{−1}— so*x*^{−1}∈*Z*(*G*).

Furthermore the center of *G* is always a normal subgroup of *G*, as it is closed under conjugation.

## Conjugacy classes and centralisers

By definition, the center is the set of elements for which the conjugacy class of each element is the element itself, i.e. ccl(g) = {g}.

The center is also the intersection of all the centralizers of each element of G. As centralizers are subgroups, this again shows that the center is a subgroup.

## Conjugation

Consider the map *f*: *G* → Aut(*G*) from *G* to the automorphism group of *G* defined by *f*(*g*) = ϕ_{g}, where ϕ_{g} is the automorphism of *G* defined by

The function *f* is a group homomorphism, and its kernel is precisely the center of *G*, and its image is called the inner automorphism group of *G*, denoted Inn(*G*). 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
*G*is all of*G*. - The center of the Heisenberg group
*G*are all matrices of the form : - The center of a nonabelian simple group is trivial.
- The center of the dihedral group D
_{n}is trivial when*n*is odd. When*n*is 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
*S*_{n}is trivial for*n*≥ 3. - The center of the alternating group
*A*_{n}is trivial for*n*≥ 4. - The center of the general linear group is the collection of scalar matrices .
- The center of the orthogonal group is .
- 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, G is abelian (and so G = Z(G), and is trivial).
- The quotient group is not isomorphic to the quaternion group .

## 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 ** ith center** of

*G*(

**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

*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**.

^{[note 2]}

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

## Notes

## External links

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