﻿ Center (algebra)

# Center (algebra)

In the mathematical field of algebra, the center of an algebra or a group refers to that subset of the considered structure, which consists of all the elements that commute with respect to all elements of the multiplication.

• 4.1 Example

## Center of a group

Is a group, whose center is the amount

The center of G is a subgroup, because x and y of Z (G ), then for each

So also in the center. Analogously, one can show that is in the center. The neutral element of the group is at the center.

### Properties

The center is abelian and a normal subgroup of G, it is even a characteristic subgroup of G, that remains fixed under every automorphism. The center is even strictly characteristic, that remains fixed under every epimorphism. If G itself is abelian, then.

The center consists of exactly the elements z of G for the conjugation, ie, the identity map is z.

### Examples

• The center of the symmetric group of degree 3 consists only of the neutral element, because:
• The dihedral group consists of the movements of the plane, which can be a fixed chosen square unchanged. These are at the center of the square to an angle of 0 °, 90 °, 180 ° and 270 °, and four mirrors on the two diagonals of the square parallel to the two means. Rotations The center of this group consists exactly of the two rotations around 0 ° and 180 °.
• The center of the multiplicative group of invertible NxN matrices with entries in the real number consisting of the real multiple of the unit matrix.

## Center of a ring

The center of a ring R consists of those elements of the ring, the commute with any other:

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

## Center of an associative algebra

The center of an associative algebra A is the commutative subalgebra

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

## Center of a Lie algebra

The center of a Lie algebra is the ( abelian ) ideal

A Lie algebra is true if and only consistent with its center 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 Lieklammer match the two center terms.
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