group center

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abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
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group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
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set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formul ...
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commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with every element of . It is denoted , from German ''wikt:Zentrum, Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, . As a subgroup, it is always characteristic subgroup, characteristic, but is not necessarily fully characteristic subgroup, fully characteristic. The quotient group, , is group isomorphism, 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 group, trivial; i.e., consists only of the identity element. The elements of the center are sometimes called central.

# As a subgroup

The center of ''G'' is always a subgroup (mathematics), 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 a normal subgroup of . Since all elements of commute, it is closed under conjugate closure, conjugation.

# Conjugacy classes and centralizers

By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., . The center is also the intersection (set theory), intersection of all the centralizer and normalizer, centralizers of each element 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 (algebra), 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: $\begin 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end$ * The center of a nonabelian group, 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 (mathematics), field , , is the collection of diagonal matrix, 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, $U\left(n\right)$ is $\left\$. * The center of the special unitary group, $\operatorname\left(n\right)$ is $\left\lbrace e^ \cdot I_n \mid \theta = \frac, k = 0, 1, \dots, n-1 \right\rbrace$. * 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 group, finite p-group is non-trivial. * If the quotient group is cyclic group, cyclic, is abelian group, abelian (and hence , so is trivial). * The center of the megaminx group is a cyclic group of 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 of (second center, third center, etc.) and is denoted . Concretely, the ()-st center are the terms 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.This union will include transfinite terms if the UCS does not stabilize at a finite stage. The total order#Chains, 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 .