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).

∘	e	b	a	a^2	a^3	ab	a^2b	a^3b
e	e	b	a	a^2	a^3	ab	a^2b	a^3b
b	b	e	a^3b	a^2b	ab	a^3	a^2	a
a	a	ab	a^2	a^3	e	a^2b	a^3b	b
a^2	a^2	a^2b	a^3	e	a	a^3b	b	ab
a^3	a^3	a^3b	e	a	a^2	b	ab	a^2b
ab	ab	a	b	a^3b	a^2b	e	a^3	a^2
a^2b	a^2b	a^2	ab	b	a^3b	a	e	a^3
a^3b	a^3b	a^3	a^2b	ab	b	a^2	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:

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\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. .

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

Z(G) = c ap _{g\in} Z _{G(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

• The center of an abelian group,, is all of .
• The center of the Heisenberg group,, is the set of matrices of the form: $$\backslash begin$$

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

• 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,

U(n) is \left\{ e ^{i\theta} ⋅ I _{n} \mid \theta \in [ 0 , 2 \pi) \right\} .

• The center of the special unitary group,

\operatorname{SU}(n) is $\backslash left\backslash lbrace\; e^\; \backslash cdot\; I\_n\; \backslash mid\; \backslash theta\; =\; \backslash frac,\; k\; =\; 0,\; 1,\; \backslash dots,\; n-1\; \backslash right\backslash rbrace$.

• 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^[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

• 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

References

• Book: Fraleigh . John B. . 2014 . A First Course in Abstract Algebra . 7 . Pearson . 978-1-292-02496-7.

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.