Center (group theory)

In abstract algebra, the center of a group is the set of elements that commute with every element of . It is denoted , from German , 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:

1. contains the identity element of , because it commutes with every element of , by definition: , where is the identity;
2. If and are in , then so is , by associativity: for each ; i.e., is closed;
3. If is in , then so is as, for all in , commutes with : .

Furthermore, the center of is always an abelian and normal subgroup of . Since all elements of commute, it is closed under conjugation.

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 : <blockquote> </blockquote>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 by

.

The 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 (and hence , so is trivial).
• The center of the Rubik's Cube group consists of two elements – the identity (i.e. the solved state) 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

Quotienting out by the center of a group yields a sequence of groups called the upper central series:

The kernel of the map is the th center of (second center, third center, etc.), 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 (equivalently, ) 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 .

See also

• Center (algebra)
• Center (ring theory)
• Centralizer and normalizer
• Conjugacy class

Notes

References

External links