Center (group theory)

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:
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
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.
The ascending chain of subgroups
stabilizes at "i" (equivalently, ) if and only if is centerless.