# 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,

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.

Furthermore, the center of *G* is always a normal subgroup of *G*. Since all elements of Z(*G*) commute, it is closed under conjugation.

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.

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 Out(*G*) of outer automorphisms, and these form the exact sequence

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

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

*i*th center^{[citation needed]}of

*G*(

**second center**,

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

^{i}(

*G*)

^{[citation needed]}. 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]}

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