Center (group theory)
| e | b | a | a2 | a3 | ab | a2b | a3b | |
| e | e | b | a | a2 | a3 | ab | a2b | a3b |
| b | b | e | a3b | a2b | ab | a3 | a2 | a |
| a | a | ab | a2 | a3 | e | a2b | a3b | b |
| a2 | a2 | a2b | a3 | e | a | a3b | b | ab |
| a3 | a3 | a3b | e | a | a2 | b | ab | a2b |
| ab | ab | a | b | a3b | a2b | e | a3 | a2 |
| a2b | a2b | a2 | ab | b | a3b | a | e | a3 |
| a3b | a3b | a3 | a2b | ab | b | a2 | a | e |
In abstract algebra, the center of a group is the set of elements that commute with every element of. It is denoted, from German Zentrum, meaning center. In set-builder notation,
The center is a normal subgroup,, and also a characteristic subgroup, but is not necessarily fully characteristic. The quotient group,, is isomorphic to the inner automorphism group,.
A group is abelian if and only if. At the other extreme, a group is said to be centerless if is trivial; i.e., consists only of the identity element.
The elements of the center are central elements.
As a subgroup
The center of G is always a subgroup of. In particular:- contains the identity element of, because it commutes with every element of, by definition:, where is the identity;
- If and are in, then so is, by associativity: for each ; i.e., is closed;
- If is in, then so is as, for all in, commutes with :.
A group homomorphism might not restrict to a homomorphism between their centers. The image elements commute with the image, but they need not commute with all of unless is surjective. Thus the center mapping is not a functor between categories Grp and Ab, since it does not induce a map of arrows.
Conjugacy classes and centralizers
By definition, an element is central whenever its conjugacy class contains only the element itself; i.e..The center is the intersection of all the centralizers of elements of :
As centralizers are subgroups, this again shows that the center is a subgroup.
Conjugation
Consider the map, from to the automorphism group of defined by, where is the automorphism of defined byThe function, is a group homomorphism, and its kernel is precisely the center of, and its image is called the inner automorphism group of, denoted. By the first isomorphism theorem we get,
The cokernel of this map is the group of outer automorphisms, and these form the exact sequence
Examples
- The center of an abelian group,, is all of.
- The center of the Heisenberg group,, is the set of matrices of the form:
- The center of a nonabelian simple group is trivial.
- The center of the dihedral group,, is trivial for odd. For even, the center consists of the identity element together with the 180° rotation of the polygon.
- The center of the quaternion group,, is.
- The center of the symmetric group,, is trivial for.
- The center of the alternating group,, is trivial for.
- The center of the general linear group over a field,, is the collection of scalar matrices,.
- The center of the orthogonal group, is.
- The center of the special orthogonal group, is the whole group when, and otherwise when n is even, and trivial when n is odd.
- The center of the unitary group, is.
- The center of the special unitary group, is.
- The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.
- Using the class equation, one can prove that the center of any non-trivial finite p-group is non-trivial.
- If the quotient group is cyclic, is abelian.
- The center of the Rubik's Cube group consists of two elements – the identity and the superflip. The center of the Pocket Cube group is trivial.
- The center of the Megaminx group has order 2, and the center of the Kilominx group is trivial.
Higher centers
The kernel of the map is the th center of , denoted. Concretely, the -st center comprises the elements that commute with all elements up to an element of the th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter.
The ascending chain of subgroups
stabilizes at i if and only if is centerless.
Examples
- For a centerless group, all higher centers are zero, which is the case of stabilization.
- By Grün's lemma, the quotient of a perfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at.