Center (group theory)
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| <math>\circ</math> | 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,
The center is a normal subgroup, <math>Z(G)\triangleleft G</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 subgroupEdit
The center of G is always a subgroup of Template:Math. In particular:
- 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;
- 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;
- 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 <math>G\to Z(G)</math> is not a functor between categories Grp and Ab, since it does not induce a map of arrows.
Conjugacy classes and centralizersEdit
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:
<math>Z(G) = \bigcap_{g\in G} Z_G(g).</math>
As centralizers are subgroups, this again shows that the center is a subgroup.
ConjugationEdit
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
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,
The cokernel of this map is the group Template:Math of outer automorphisms, and these form the exact sequence
ExamplesEdit
- 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: <math display="block"> \begin{pmatrix}
1 & 0 & z\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}</math>
- 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, <math>U(n)</math> is <math>\left\{ e^{i\theta} \cdot I_n \mid \theta \in [0, 2\pi) \right\}</math>.
- The center of the special unitary group, <math>\operatorname{SU}(n)</math> is <math display="inline">\left\lbrace e^{i\theta} \cdot I_n \mid \theta = \frac{2k\pi}{n}, k = 0, 1, \dots, n-1 \right\rbrace </math>.
- 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 centersEdit
Quotienting out by the center of a group yields a sequence of groups called the upper central series:
The kernel of the map Template:Math is the Template:Mathth center<ref>Template:Cite journal</ref> of Template:Math (second center, third center, etc.), denoted Template:Math.<ref>Template:Cite journal</ref> 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.<ref group="note">This union will include transfinite terms if the UCS does not stabilize at a finite stage.</ref>
The ascending chain of subgroups
stabilizes at i (equivalently, Template:Math) if and only if Template:Math is centerless.
ExamplesEdit
- 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.