# Center (group theory)

# 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−1*commutes with*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

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* → *Gi* is the * i* of

*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**.

^{[1]}

The ascending chain of subgroups

- 1 ≤ Z(

*G*) ≤ Z

^{2}(

*G*) ≤ ⋯

stabilizes at *i* (equivalently, Z*i*(*G*) = Zi+1(*G*)) if and only if *G**i* is centerless.

Examples

For a centerless group, all higher centers are zero, which is the case Z0(

*G*) = Z1(*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 Z1(

*G*) = Z2(*G*).

See also

Center (algebra)

center

Centralizer and normalizer

Conjugacy class