Center (group theory)

Cayley table for D₄ showing elements of the center, {e, a²}, 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).

∘	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 Template:Math is the set of elements that commute with every element of Template:Math. It is denoted Template:Math, from German Zentrum, meaning center. In set-builder notation,

Template:Math.

The center is a normal subgroup, Template:Math, and also a characteristic subgroup, but is not necessarily fully characteristic. The quotient group, Template:Math, is isomorphic to the inner automorphism group, Template:Math.

A group Template:Math is abelian if and only if Template:Math. At the other extreme, a group is said to be centerless if Template:Math 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 Template:Math. In particular:

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

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

A group homomorphism Template:Math might not restrict to a homomorphism between their centers. The image elements Template:Math commute with the image Template:Math, but they need not commute with all of Template:Math unless Template:Math is surjective. Thus the center mapping $G\to Z(G)$ 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. Template:Math.

The center is the intersection of all the centralizers of elements of Template:Math:

$Z(G)=\bigcap _{g\in G}{Z}_G(g).$

As centralizers are subgroups, this again shows that the center is a subgroup.

Conjugation

Consider the map Template:Math, from Template:Math to the automorphism group of Template:Math defined by Template:Math, where Template:Math is the automorphism of Template:Math defined by

Template:Math.

The function, Template:Math is a group homomorphism, and its kernel is precisely the center of Template:Math, and its image is called the inner automorphism group of Template:Math, denoted Template:Math. By the first isomorphism theorem we get,

Template:Math.

The cokernel of this map is the group Template:Math of outer automorphisms, and these form the exact sequence

Template:Math.

Examples

• The center of an abelian group, Template:Math, is all of Template:Math.
• The center of the Heisenberg group, Template:Math, is the set of matrices of the form: $$\left(\begin{array}{ccc}1& 0& z\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$
• The center of a nonabelian simple group is trivial.
• The center of the dihedral group, Template:Math, is trivial for odd Template:Math. For even Template:Math, the center consists of the identity element together with the 180° rotation of the polygon.
• The center of the quaternion group, Template:Math, is Template:Math.
• The center of the symmetric group, Template:Math, is trivial for Template:Math.
• The center of the alternating group, Template:Math, is trivial for Template:Math.
• The center of the general linear group over a field Template:Math, Template:Math, is the collection of scalar matrices, Template:Math.
• The center of the orthogonal group, Template:Math is Template:Math.
• The center of the special orthogonal group, Template:Math is the whole group when Template:Math, and otherwise Template:Math when n is even, and trivial when n is odd.
• The center of the unitary group, $U(n)$ is $\{e^{i\theta }\cdot I_n\mid \theta \in [0,2\pi )\}$.
• The center of the special unitary group, $\mathrm{SU}(n)$ is $\{e^{i\theta }\cdot I_n\mid \theta =\frac{2k\pi }{n},k=0,1,\dots ,n-1\}$.
• 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 Template:Math is cyclic, Template:Math is abelian (and hence Template:Math, so Template:Math 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:

Template:Math

The kernel of the map Template:Math is the Template:Mathth center of Template:Math (second center, third center, etc.), denoted Template:Math. Concretely, the (Template:Math)-st center comprises the elements that commute with all elements up to an element of the Template:Mathth 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

Template:Math

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

Examples

• For a centerless group, all higher centers are zero, which is the case Template:Math 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 Template:Math.

See also

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

References

• Template:Cite book

External links

• Template:Springer