Center (algebra)

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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


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 .


  • 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


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.


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


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