Center (group theory)

Cayley table of Dih₄, the dihedral group of order 8.Template:Paragraph break The center is {0,7}: The row starting with 7 is the transpose of the column starting with 7. The entries 7 are symmetric to the main diagonal. (Only for the identity element is this true in all groups.)

In abstract algebra, the center of a group, Template:Math, is the set of elements that commute with every element of Template:Math. It is denoted Template:Math, from German Zentrum, meaning center. In set-builder notation,

Template:Math.

The center is a normal subgroup, Template:Math. As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, Template:Math, is isomorphic to the inner automorphism group, Template:Math.

A group Template:Math is abelian if and only if Template:Math. At the other extreme, a group is said to be centerless if Template:Math 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 Template:Math. In particular:

1. Template:Math contains the identity element of Template:Math, because it commutes with every element of Template:Math, by definition: Template:Math, where Template:Math is the identity;
2. If Template:Math and Template:Math are in Template:Math, then so is Template:Math, by associativity: Template:Math for each Template:Math; i.e., Template:Math is closed;
3. If Template:Math is in Template:Math, then so is Template:Math as, for all Template:Math in Template:Math, Template:Math commutes with Template:Math: Template:Math.

Furthermore, the center of Template:Math is always a normal subgroup of Template:Math. Since all elements of Template:Math 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., Template:Math}.

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

Conjugation

Consider the map, Template:Math, from Template:Math to the automorphism group of Template:Math defined by Template:Math, where Template:Math is the automorphism of Template:Math defined by

Template:Math.

The function, Template:Math is a group homomorphism, and its kernel is precisely the center of Template:Math, and its image is called the inner automorphism group of Template:Math, denoted Template:Math. By the first isomorphism theorem we get,

Template:Math.

The cokernel of this map is the group Template:Math of outer automorphisms, and these form the exact sequence

Template:Math.

Higher centers

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

Template:Math

The kernel of the map, Template:Math is the Template:Mathth center of Template:Math (second center, third center, etc.), and is denoted Template:Math. Concretely, the (Template:Math)-st center are the terms that commute with all elements up to an element of the Template:Mathth 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

Template:Math

stabilizes at i (equivalently, Template:Math) if and only if Template:Math is centerless.

Examples

• For a centerless group, all higher centers are zero, which is the case Template:Math 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 Template:Math.

See also

• Center (algebra)
• center
• Centralizer and normalizer
• Conjugacy class

References

• Template:Cite book

External links

• Template:Springer