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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