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 | a2 | a3 | ab | a2b | a3b |
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e | e | b | a | a2 | a3 | ab | a2b | a3b |
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b | b | e | a3b | a2b | ab | a3 | a2 | a |
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a | a | ab | a2 | a3 | e | a2b | a3b | b |
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a2 | a2 | a2b | a3 | e | a | a3b | b | ab |
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a3 | a3 | a3b | e | a | a2 | b | ab | a2b |
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ab | ab | a | b | a3b | a2b | e | a3 | a2 |
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a2b | a2b | a2 | ab | b | a3b | a | e | a3 |
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a3b | a3b | a3 | a2b | ab | b | a2 | a | e | |
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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:
- contains the identity element of, because it commutes with every element of, by definition:, where is the identity;
- If and are in, then so is, by associativity: for each ; i.e., is closed;
- 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 :
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
- 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
\left\{ei\theta ⋅ In\mid\theta\in[0,2\pi)\right\}
.
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[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
References
- Book: Fraleigh . John B. . 2014 . A First Course in Abstract Algebra . 7 . Pearson . 978-1-292-02496-7.
Notes and References
- 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.
- 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.
- This union will include transfinite terms if the UCS does not stabilize at a finite stage.