Center (group Theory)
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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the center of a
group A group is a number of persons 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 ide ...
, , is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of elements that
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 '' Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, . As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
, , is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
group, . A group is abelian if and only if . At the other extreme, a group is said to be centerless if is
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
; i.e., consists only of the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. The elements of the center are sometimes called central.


As a subgroup

The center of ''G'' is always a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of . In particular: # contains the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
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
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
. Note that a homomorphism between groups generally does not restrict to a homomorphism between their centers. Although commutes with , unless is surjective need not commute with all of and therefore need not be a subset of . Put another way, there is no "center" functor between categories Grp and Ab. Even though we can map objects, we cannot map arrows.


Conjugacy classes and centralizers

By definition, the center is the set of elements for which the
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
of each element is the element itself; i.e., . The center is also the intersection of all the 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 In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of defined by , where is the automorphism of defined by :. The function, is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
, and its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
is precisely the center of , and its image is called the
inner automorphism group In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group its ...
of , denoted . By the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
we get, :. The
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of this map is the group of
outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
s, and these form the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
:.


Examples

* The center of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, , is all of . * The center of the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
, , is the set of matrices of the form: \begin 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end * The center of a nonabelian
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
is trivial. * The center of the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
, , is trivial for odd . For even , the center consists of the identity element together with the 180° rotation of the
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
. * The center of the
quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 ...
, , is . * The center of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
, , is trivial for . * The center of the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
, , is trivial for . * The center of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, , is the collection of
scalar matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
, . * The center of the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
, 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 In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
, U(n) is \left\. * The center of the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, \operatorname(n) 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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s is the multiplicative group of non-zero
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. * Using the
class equation In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
, one can prove that the center of any non-trivial
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
p-group In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ...
is non-trivial. * If the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
is
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
, is abelian (and hence , so is trivial). * The center of the
megaminx The Megaminx or Mégaminx (, ) is a dodecahedron-shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces of the Rubik's Cube. History The Megaminx, or Magic Dodecahedron ...
group is a cyclic group of order 2, and the center of the
kilominx The Megaminx or Mégaminx (, ) is a dodecahedron-shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces of the Rubik's Cube. History The Megaminx, or Magic Dodecahedron, ...
group is trivial.


Higher centers

Quotienting out by the center of a group yields a sequence of groups called the
upper central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central ...
: : 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 In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qu ...
by
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...
; the union of all the higher centers is called the
hypercenter In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...
.This union will include transfinite terms if the UCS does not stabilize at a finite stage. The ascending chain of subgroups : stabilizes at ''i'' (equivalently, )
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
is centerless.


Examples

* For a centerless group, all higher centers are zero, which is the case of stabilization. * By
Grün's lemma In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the univers ...
, the quotient of a
perfect group In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universa ...
by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at .


See also

*
Center (algebra) The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. * The center of a group ''G'' consists of all those elements ''x'' in ''G'' such that ''xg'' = ...
*
Center (ring theory) In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that ''xy = yx'' for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(R); "Z" stands for the German word ''Zentrum'', meaning "c ...
*
Centralizer and normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
*
Conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...


Notes


References

*


External links

* {{springer, title=Centre of a group, id=p/c021250 Group theory Functional subgroups