# Center (algebra)

In the mathematical subfield of algebra , the **center of** an algebra or a group denotes that subset of the structure under consideration, which consists of all the elements that commute with all the elements in terms of multiplication .

## Center of a group

If there is a group, its center is the set

### properties

The center of is a subgroup , because are and off , then applies to each

so is also in the center. Analogously one shows that in the center lies:

- .

The neutral element of the group is always in the center: .

The center is Abelian and a normal divisor of , it is even a characteristic subgroup of . is Abelian if and only if .

The center consists of exactly those elements of for which the conjugation with , that is , the identical mapping. Thus, the center can also be defined as a special case of the centralizer . It applies .

### Examples

- The center of the symmetrical group of degree 3 consists only of the neutral element , because:

- The dihedral group consists of the movements of the plane that leave a fixed square unchanged. These are the rotations around the center of the square by angles of 0 °, 90 °, 180 ° and 270 °, as well as four reflections on the two diagonals and the two central parallels of the square. The center of this group consists exactly of the two rotations of 0 ° and 180 °.
- The center of the multiplicative group of invertible
*n*×*n*- matrices with entries in the real numbers consists of the real multiple of the unit matrix .

## Center of a ring

The **center of** a ring consists of those elements of the ring that commute with all others:

The center is a commutative subring of . A ring coincides with its center if and only if it is commutative.

## Center of an associative algebra

The **center of** an associative algebra is the commutative sub-
algebra

An algebra coincides with its center if and only if it is commutative .

## Center of a Lie algebra

### definition

The **center of** a Lie algebra is the (Abelian) ideal

- ,

where the Lie bracket denotes the multiplication in . A Lie algebra coincides with its center if and only if it is Abelian.

### example

- The center of the general linear group consists of the scalar multiples of the identity matrix

- .

- For an associative algebra with the commutator as a Lie bracket, the two center terms agree.

## literature

- Kurt Meyberg:
*Algebra - Part 1*. Hanser 1980, ISBN 3-446-13079-9 , p. 36

## Web links

- Center in various algebraic structures at PlanetMath (English)