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 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:
Center of an associative algebra
An algebra coincides with its center if and only if it is commutative .
Center of a Lie algebra
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.
- Kurt Meyberg: Algebra - Part 1 . Hanser 1980, ISBN 3-446-13079-9 , p. 36