# Center (group theory)

In abstract algebra, the **center** of a group, *G*, is the set of elements that commute with every element of *G*. It is denoted Z(*G*), from German *Zentrum,* meaning *center*. In set-builder notation,

- Z(
*G*) = {*z*∈*G*∣ ∀*g*∈*G*,*zg*=*gz*} .

The center is a normal subgroup, Z(*G*) ⊲ *G*. As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, *G* / Z(*G*), is isomorphic to the inner automorphism group, Inn(*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**.

## As a subgroup

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

- Z(
*G*) contains the identity element of*G*, because it commutes with every element of*g*, by definition:*eg*=*g*=*ge*, where*e*is the identity; - If
*x*and*y*are in Z(*G*), then so is*xy*, by associativity: (*xy*)*g*=*x*(*yg*) =*x*(*gy*) = (*xg*)*y*= (*gx*)*y*=*g*(*xy*) for each*g*∈*G*; i.e., Z(*G*) is closed; - If
*x*is in Z(*G*), then so is*x*^{−1}as, for all*g*in*G*,*x*commutes with^{−1}*g*: (*gx*=*xg*) ⇒ (*x*^{−1}*gxx*^{−1}=*x*^{−1}*xgx*^{−1}) ⇒ (*x*^{−1}*g*=*gx*^{−1}).

Furthermore, the center of *G* is always a normal subgroup of *G*. Since all elements of Z(*G*) 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., Cl(*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

*f*(*g*)(*h*) =*ϕ*_{g}(*h*) =*ghg*^{−1}.

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,

*G*/Z(*G*) ≃ Inn(*G*).

The cokernel of this map is the group Out(*G*) of outer automorphisms, and these form the exact sequence

- 1 ⟶ Z(
*G*) ⟶*G*⟶ Aut(*G*) ⟶ Out(*G*) ⟶ 1.

## Examples

- The center of an abelian group,
*G*, is all of*G*. - The center of the Heisenberg group,
*H*, is the set of 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, Q
_{8}= {1, −1, i, −i, j, −j, k, −k} , is {1, −1} . - 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, GL
_{n}(F), is the collection of scalar matrices, {sI_{n}∣ s ∈ F \ {0}}. - The center of the orthogonal group, O(
*n*, F) is {I_{n}, −I_{n}}. - The center of the special orthogonal group, SO(
*n*) is {I_{n}, −I_{n}} when*n*is even, and trivial when*n*is odd. - The center of the unitary group, is .
- The center of the special unitary 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,
*G*/Z(*G*), is cyclic,*G*is abelian (and so*G*= Z(*G*), and*G*/Z(*G*) is trivial).

## Higher centers

Quotienting out by the center of a group yields a sequence of groups called the **upper central series**:

- (
*G*_{0}=*G*) ⟶ (*G*_{1}=*G*_{0}/Z(*G*_{0})) ⟶ (*G*_{2}=*G*_{1}/Z(*G*_{1})) ⟶ ⋯

The kernel of the map, *G* → *G _{i}* is the

**of**

*i*th center*G*(

**second center**,

**third center**, etc.), and is denoted Z

^{i}(

*G*). Concretely, the (

*i*+ 1)-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 1]

The ascending chain of subgroups

- 1 ≤ Z(
*G*) ≤ Z^{2}(*G*) ≤ ⋯

stabilizes at *i* (equivalently, Z^{i}(*G*) = Z^{i+1}(*G*)) if and only if *G*_{i} is centerless.

### Examples

- For a centerless group, all higher centers are zero, which is the case Z
^{0}(*G*) = Z^{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 Z
^{1}(*G*) = Z^{2}(*G*).

## Notes

- This union will include transfinite terms if the UCS does not stabilize at a finite stage.

## References

- Fraleigh, John B. (2014).
*A First Course in Abstract Algebra*(7 ed.). Pearson. ISBN 978-1-292-02496-7.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Centre of a group",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4