Center (group theory) explained

Cayley table for D4 showing elements of the center,, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are transposes of each other).
ebaa2a3aba2ba3b
eebaa2a3aba2ba3b
bbea3ba2baba3a2a
aaaba2a3ea2ba3bb
a2a2a2ba3eaa3bbab
a3a3a3beaa2baba2b
abababa3ba2bea3a2
a2ba2ba2abba3baea3
a3ba3ba3a2babba2ae
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:

  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

G\toZ(G)

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 :

Z(G)=capg\inZG(g).

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

1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end

U(n)

is

\left\{ei\thetaIn\mid\theta\in[0,2\pi)\right\}

.

\operatorname{SU}(n)

is \left\lbrace e^ \cdot I_n \mid \theta = \frac, k = 0, 1, \dots, n-1 \right\rbrace .

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[1] of (second center, third center, etc.), denoted .[2] 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.[3]

The ascending chain of subgroups

stabilizes at i (equivalently,) if and only if is centerless.

Examples

See also

References

Notes and References

  1. Ellis . Graham . 1998-02-01 . On groups with a finite nilpotent upper central quotient . Archiv der Mathematik . en . 70 . 2 . 89–96 . 10.1007/s000130050169 . 1420-8938.
  2. Ellis . Graham . 1998-02-01 . On groups with a finite nilpotent upper central quotient . Archiv der Mathematik . en . 70 . 2 . 89–96 . 10.1007/s000130050169 . 1420-8938.
  3. This union will include transfinite terms if the UCS does not stabilize at a finite stage.