The possible rearrangements of Rubik's Cube form a group, called the Rubik's Cube group. The Rubik's Cube is a Mechanical puzzle invented in 1974 by Hungarian Sculptor and Professor of Architecture Ernő Rubik The Rubik's Cube provides a tangible representation of a mathematical group.

Groups

Group theory

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A group is one of the fundamental objects of study in the field of mathematics known as abstract algebra, and more specifically group theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. ^[1][2] Groups are sets with an additional operation which combines any two elements to a third one, subject to certain conditions familiar from number systems such as the integers, or the rational numbers with addition as the group operation, as well as the non-zero rational numbers with multiplication. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Addition is the mathematical process of putting things together

Groups often occur in the guise of symmetry groups of geometrical objects. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Groups, in particular Lie groups such as groups of matrices, are essential abstractions in branches of physics involving symmetry principles. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Their ability to represent geometric transformations also finds applications in chemistry. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Groups are heavily influential in other mathematical domains. Important algebraic structures such as rings and fields can be defined concisely in terms of groups. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Historically, groups are rooted in several parallel origins such as permutation groups, culminating in a notion that allows to investigate properties of groups in a general and abstract setting. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation Beyond direct implications of the group axioms, basic techniques include studying groups related to a given one (such as sub- or quotient groups) or decomposing groups into simpler parts. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning A particularly ample theory has been developed for finite groups, coming to the climax of the classification of finite simple groups. In Mathematics, a finite group is a group which has finitely many elements The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

Definition and illustration

A group (G, •) is a set G with a binary operation • on G that satisfies the following four axioms:^[3]

1. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject	Closure. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set	For all a, b in G, the result of a • b is also in G.
2.	Associativity. In Mathematics, associativity is a property that a Binary operation can have	For all a, b and c in G, (a • b) • c = a • (b • c).
3.	Identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that	There exists an element e in G such that for all a in G, e • a = a • e = a.
4.	Inverse element. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to	For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

First example: the integers

The first example of a group is the set of integers Z = {. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . }. Under the usual addition operation "+" they form what is probably the most familiar group. Addition is the mathematical process of putting things together It is denoted by (Z, +). The group axioms can be thought of as being modeled on the properties of the integers together with the addition operation. The abstract group axioms reduce to statements about numbers, in this case:

1.	Closure.	For any two integers a, b, the sum a + b is also an integer.
2.	Associativity.	For all integers a, b and c, (a + b) + c = a + (b + c).
3.	Identity element.	If a is any integer, then 0 + a = a + 0 = a. Thus 0 is the (additive) identity.
4.	Inverse element.	For each a in Z, b = −a is an integer and satisfies a + b = b + a = 0. Thus −a is the (additive) inverse of the integer a.

Worked example: a symmetry group

id (keeping it as is) r1 (rotation by 90°) r2 (rotation by 180°) r3 (left rotation by 90°) fv (vertical flip) fh (horizontal flip) fd (diagonal flip) fc (counter-diagonal flip) Elements of the symmetry group. The vertices are colored only to visualize the operations.

group table • id r1 r2 r3 fv fh fd fc id id r1 r2 r3 fv fh fd fc r1 r1 r2 r3 id fc fd fv fh r2 r2 r3 id r1 fh fv fd fc r3 r3 id r1 r2 fd fc fh fv fv fv fd fh fc id r2 r1 r3 fh fh fc fv fd r2 id r3 r1 fd fd fh fc fv r3 r1 id r2 fc fc fv fd fh r1 r3 r2 id The elements id, r1, r2, and r3 form a subgroup (highlighted in red). A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a

Group theory and the notion of a group concern much more general entities than numbers. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. The following illustrates the meaning of the group axioms for the dihedral group of symmetries of the square. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides ^[4] The elements of the group are operations which keep the shape of the square unchanged. The operations are:

• Three rotations r₁, r₂ and r₃ (rotating the square by 90°, 180°, and 270° respectively). A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation
• Reflections about the vertical and horizontal middle line (f_h and f_v), or through the two diagonals (f_d and f_c). In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line
• The identity operation id leaving everything unchanged.

In this example group, the axioms can be understood as follows:

1. The closure axiom demands that any two symmetries can be composed. In Mathematics, a composite function represents the application of one function to the results of another This is indeed the case—for any two symmetries a and b, we can first perform a and then b and the result will still be a symmetry, written symbolically from right to left ("perform the symmetry b after performing the symmetry a") as:

   b • a.

   For example, rotating by 90° left (r₃) and then flipping horizontally (f_h) is the same as performing a reflection along the diagonal (f_d). Using the above symbols, we have (highlighted in blue in the group table):

   f_h • r₃ = f_d.

2. The associativity constraint is the natural axiom to impose in order to make composing more than two symmetries well-behaved: given three elements a, b and c of G, there are two possible ways of computing "a after b after c". The requirement:

   (a • b) • c = a • (b • c)

   means that composing a after b, and calling this symmetry x, then x after c is the same as a after y, where y in turn is applying b after c. For example, we check (f_d • f_v) • r₂ = f_d • (f_v • r₂) using the group table at the right:

   (f_d • f_v) • r₂ = r₃ • r₂ = r₁, which equals

   f_d • (f_v • r₂) = f_d • f_h = r₁. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a

3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form:

   id • a = a, and

   a • id = a.

4. An inverse element undoes the operation of some other element. In the symmetry group example, every symmetry can be undone: each of the identity id, the flips f_h, f_v, f_d, f_c and the 180° rotation r₂ is its own inverse, because performing each one twice brings the square back to its original orientation. Each of the 90° rotations r₃ and r₁ is each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols, for example:

   f_h • f_h = id,

   r₃ • r₁ = r₁ • r₃ = id.

[edit] History

Main article: History of group theory

Historically, the group concept has evolved in several parallel threads. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. ^[5][6][7] One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Galois, extending prior work of Ruffini and Lagrange, introduced groups of solutions of such equations, thus giving a criterion for the solvability of such equations. Paolo Ruffini ( September 22, 1765 – May 9, 1822) was an Italian Mathematician and Philosopher. In Mathematics, a Galois group is a group associated with a certain type of Field extension. ^[8] Cauchy pushed the theory of permutation groups further. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation Cayley's On the theory of groups, as depending on the symbolic equation θⁿ = 1 gives the first abstract definition of finite groups. In Mathematics, a finite group is a group which has finitely many elements

Secondly, the relation of groups to geometry was initiated by Klein's 1872 Erlangen program. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a discrete group is a group G equipped with the Discrete topology. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician.

The third root of group theory was number theory: certain abelian group structures had been implicitly used in number-theoretical work by Gauss, and more explicitly by Kronecker. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued ^[9] Early attempts to prove Fermat's last theorem were led to a climax by Kummer by introducing groups describing factorization into prime numbers. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

The convergence of these various sources into a uniform theory of groups started with Jordan's Traité des substitutions et des équations algébriques (1870) and von Dyck (1882) who first defined a group in the full abstract sense of this article. Marie Ennemond Camille Jordan ( January 5 1838 &ndash January 22 1922) was a French Mathematician, known both for his foundational Year 1870 ( MDCCCLXX) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common Walther Franz Anton von Dyck ( December 6, 1856 - November 5, 1934) was a German Mathematician. Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common The early 20th century's group theory encompassed roughly the content of the basic concepts (see below). Group theory subsequently grew both in depth and in breadth, branching out into areas such as algebraic groups, group extensions and representation theory. In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of The latter was crucial for the success of the classification of finite simple groups in 1982, a major accomplishment of contemporary group theory. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. Year 1982 ( MCMLXXXII) was a Common year starting on Friday (link displays the 1982 Gregorian calendar) ^[10]

First consequences of the group axioms

Elementary group theory is concerned with basic facts about general groups, as opposed for example to the more involved study of groups via their representations. In Mathematics, a group G,*> is defined as a set G and a Binary operation on G, called product and denoted In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of ^[11] These facts are usually direct consequences of the group definition — obtained by invoking the axioms a few times — and are often used in group theory without explicit reference to the corresponding statement. ^[12]

For example, repeated applications of the associativity axiom show that the unambiguity of

a • b • c = (a • b) • c = a • (b • c)

generalizes to more than three factors. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Therefore parentheses are usually omitted in such expressions.

Uniqueness of identity element and inverses

Though the uniqueness of the identity is not required by the group axioms, it is a consequence of them. Therefore it is customary to speak of the identity, and the inverse of a. ^[13]

The following proof of this fact shows the flavor of elementary group theory: suppose both e and f are identity elements. Then

e = e • f = f,

because e is a (left) identity element and f is a (right) identity element. Hence the two identities necessarily agree. Similarly, suppose given two inverses l and r of a fixed element a. Then

l = l • e = l • (a • r) = (l • a) • r = e • r = r.

Moreover, in a group, knowing only that b • a = e (or a • b = e) suffices to conclude that b is the inverse element of a. ^[14]

The inverse of a product is the product of the inverses in the opposite order: (a • b)⁻¹ = b⁻¹ • a⁻¹. The identity (a • b) • (b⁻¹ • a⁻¹) = e then suffices to prove that b⁻¹ • a⁻¹ is the inverse of a • b.

(a • b) • (b−1 • a−1)	=	((a • b) • b−1 ) • a−1	(associativity)
	=	(a • (b • b−1)) • a−1	(associativity)
	=	(a • e) • a−1	(definition of inverse)
	=	a • a−1	(definition of identity element)
	=	e	(definition of inverse)

Division

In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent In fact, right multiplication of the equation by a⁻¹ gives the solution x = x • a • a⁻¹ = b • a⁻¹. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a⁻¹ • b. In general, x and y need not agree.

Variants of the definition

Some definitions of a group use seemingly weaker conditions for identity and inverse elements. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Mathematics, especially in Category theory and Homotopy theory In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships For instance, the axioms may be weakened to assert only the existence of a left identity and a left inverse for every element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that A left inverse in Mathematics may refer to A left Inverse element with respect to a binary operation on a set A left inverse function Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one above. ^[15]

Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on G. Many authors therefore omit this axiom. ^[16]

In abstract algebra, more general structures arise by relaxing some of the axioms defining a group, shown in the table. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules ^[17][18][19] For example, eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation The integers under multiplication (Z, •) are an example (see below). Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. In Mathematics, especially in Category theory and Homotopy theory They arise in the study of more involved kinds of symmetries, often in topological and analytical structures, e. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Analysis (from Greek ἀνάλυσις, "a breaking up" is the process of breaking a complex topic or substance into smaller parts to gain a g. the fundamental groupoid. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

Notations

The notation for groups often depends on the context and the nature of the group operation. In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself There is a tendency to denote abelian groups additively, whereas non-abelian groups are often written multiplicatively. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the ^[20] In many situations, there is only one possible (or reasonable) group operation on a given set, therefore it is very common to drop the operation symbol and leave it to the reader to know the context and the group operation. For example the groups (Z_n, +) and (F_q*, ×), the multiplicative group of nonzero elements in the finite field F_q are commonly denoted Z_n and F_q*, since only one of the two ring operations makes these sets into a group. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real ^[21]

Basic concepts

The arsenal of basic group theory comprises various methods to manipulate groups. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally A group ( G, • is a set G closed under a Binary operation • satisfying the following 3 Axioms: The structure of groups can be understood by breaking them into pieces called subgroups and quotient groups. Combining them into larger groups yields direct and semidirect products. An equally important technique, fundamental to the orientation of group theory, is comparing groups using homomorphisms. A particularly well-understood class of groups are the abelian groups. These basic concepts form the standard introduction to groups. ^[22]

Subgroups

Main article: Subgroup

Informally, a subgroup is a group contained in a bigger one. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of More precisely, a subset H of G is called a subgroup if the restriction of • to H is a group operation on H. ^[23] In other words, the identity element of G is contained in H, and whenever g and h are in H, then so is g • h and g⁻¹. In the example above, the rotations {id, r₁, r₂, and r₃} constitute a subgroup (highlighted in red in the group table above): any two rotations composed are still a rotation, and a rotation can be undone by (i. e. is inverse to) the rotation in the opposite direction. It can be read off the group table above, and is indeed a general principle that knowing the subgroups of a group is important to understand the structure of the group in question. The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g⁻¹h ∈ H for all elements g, h ∈ H. In Abstract Algebra, the one-step subgroup test is a theorem that states that for any group a Subset of that group is itself a group if the inverse of

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is also the smallest subgroup containing S. In the introductory example above, the subgroup generated by r₂ and f_v consists of these two elements, the identity element id and f_h = f_v • r₂. Again, this is subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, the same elements) yields an element of this subgroup.

A subgroup H defines a set of left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In symbolic terms, the left and right coset of H containing g are

gH = {gh, h ∈ H} and Hg = {hg, h ∈ H}, respectively. ^[24]

The set of left cosets of H forms a partition of the elements of G; that is, two left cosets are either equal or have an empty intersection. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently ^[25] The same holds true of the right cosets of H. Left and right cosets of H may or may not be equal. If they are, i. e. for all g in G, gH = Hg, then H is said to be a normal subgroup. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. One may then speak simply of the set of cosets of N.

Quotient groups

Main article: Quotient group

Quotient groups, also known as factor groups, treat the cosets of a normal subgroup as a group. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G ^[26] The set of cosets of N may be equipped with an operation (sometimes called coset multiplication, or coset addition) to form a new group, called the quotient group G/N. The operation between the cosets behaves in the nicest way possible: (Ng) • (Nh) = N(gh) for all g and h in G. The coset N itself serves as the identity in this group, and the inverse of Ng in the quotient group is (Ng)⁻¹ = N(g⁻¹). ^[27]

Quotient and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. In Mathematics, one method of defining a group is by a presentation. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the The introductory dihedral group, for example, is presented by two generators r and f (for example, r = r₁, the right rotation and f = f_v the vertical (or any other) flip), together with the relations

r ⁴ = f ² = (rf)² = 1. ^[28]

A presentation of a group can also be used to construct the Cayley graph, a graphical device showing certain features of discrete groups. In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. In Mathematics, a discrete group is a group G equipped with the Discrete topology.

Taking subgroups and quotients of a given group G tends to reduce the size of G. ^[29] Several group constructions reverse this direction, i. e. given two groups, one constructs bigger groups, such as the direct product G×H of the two. In Mathematics, one can often define a direct product of objectsalready known giving a new one (Here, "product" has a slightly different meaning than the product of elements in a group. ) It consists of pairs (g, h), g in G and h in H, with the group operation

(g₁, h₁) • (g₂, h₂) = (g₁ • g₂, h₁ • h₂). In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry ^[30]

A further generalization of the direct product of two groups is the semidirect product; it allows for the twisting of the group operation on one factor. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can The group of symmetries of the square (described above) is a semidirect product of Z₄ (the subgroup consisting of rotations) with Z₂ (generated by a reflection).

Group homomorphisms

Main article: Group homomorphism

Group homomorphisms (from Greek μορφη–structure) are mappings that preserve the structure of the groups in question. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c In Mathematics and related technical fields the term map or mapping is often a Synonym for function. The structure being determined by the group operation, this is made formal by requiring

a(g • k) = a(g) • a(k).

for a map a: G → H and any two elements g, k in G. This requirement ensures that a(1_G) = 1_H, and also a(g)⁻¹ = a(g⁻¹) for all g in G, so the additional data from the group axioms are respected, as well. ^[31]

Two groups G and H are called isomorphic if there exist group homomorphisms a between G and H and b: H → G, such that applying the two maps one after another (in the two possible ways) equal the identity function of G and H, respectively, i. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that e. a(b(h)) = h, and b(a(g)) = g for any g in G and h in H. Two isomorphic groups as above carry practically the same information. For example, proving that g • g = 1 for some element g of G is equivalent to proving that a(g) • a(g) = 1, because the applying a to the first equality yields the second, and applying b to the second gives back the first. This method of dissociating the group from its concrete nature, and focussing instead on the abstract properties is a steadily recurring and deeply impacting theme in algebra, and many other mathematical domains, as well. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. The category of groups is an abstract framework containing groups and group homomorphisms. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such

For any group homomorphism a: G → H, the kernel ker a = {g in G : a(g) = 1_H} is the set of elements in G which are mapped to the identity in H. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism The kernel and image a(G) = {a(g), g ∈ G} of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The First Isomorphism Theorem states that the image of a group homomorphism, a(G) is isomorphic to the quotient group G/ker a. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage ^[32]

Abelian groups

Main article: Abelian group

A group G is said to be abelian (in honor of Niels Henrik Abel), or commutative, if the operation satisfies the commutative law

a • b = b • a. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation In Mathematics, commutativity is the ability to change the order of something without changing the end result ^[33]

for all group elements a and b. If not, the group is called non-abelian or non-commutative. As the realm of abelian groups is particularly well-understood, many group-related notions, such as the center, or commutators describe the extent to which a given group is not abelian. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. ^[34]

The group of symmetries of the square (discussed above) is non-abelian, because r₁ • f_v = f_c, which is not equal to f_v • r₁ = f_d. The subgroup {id, r₁, r₂, r₃} consisting of the rotations, as well as the quotient with respect to this subgroup, however, are abelian. This fact is reflected in the semi-direct product structure of this group (see above).

Cyclic groups

The 6^th complex roots of unity form a cyclic group. ζ is a primitive element, but ζ² is not, because the odd powers of ζ are not a power of ζ².

Main article: Cyclic group

Cyclic groups are groups whose elements may be generated by successive compositions of the group operation applied to a single element a of that group. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Mathematics, a composite function represents the application of one function to the results of another ^[35] In multiplicative notation, the group therefore consists of the powers

. . . , a⁻³, a⁻², a⁻¹, a⁰ = e, a, a², a³, . . . ,

where a² means a•a, and a⁻³ stands for a⁻¹•a⁻¹•a⁻¹=(a•a•a)⁻¹ etc. ^[36] Such an element a is called a generator or a primitive element of the group. In Modular arithmetic, a branch of Number theory, a primitive root modulo n is any number g with the property that any number Coprime

Any cyclic group is abelian. It may or may not be finite. If so, the group is isomorphic to Z_n, where n is the smallest integer such that n • a = 0. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an ^[21] The eponym is the group of n-th complex roots of unity, given by complex numbers ω satisfying ωⁿ = 1. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted ^[37] An infinite cyclic group is isomorphic to (Z, +). ^[38]

In any group G, the powers of any group element a and their inverses form a subgroup of G, called the cyclic subgroup generated by a.

Examples and applications

The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps In

Main articles: Examples of groups and Applications of group theory

Examples and applications of groups abound. Some elementary examples of groups in Mathematics are given on Group (mathematics. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. A starting point is the group Z of integers (with addition as group operation), introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak These groups are predecessors of important constructions in abstract algebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

Groups are also applied in many other mathematical areas. A major theme in contemporary mathematics is to study given objects by associating groups to them. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories For example, Poincaré founded what is now called algebraic topology by introducing the fundamental and higher homotopy groups. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional By means of this connection, topological properties translate into group-theoretic properties. This is a glossary of some terms used in the branch of Mathematics known as Topology. ^[39] In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. ^[40] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and In Group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated Further branches crucially applying groups include number theory and algebraic geometry. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Applications of group theory are by no means restricted to mathematics. Other sciences such as physics, chemistry or computer science benefit from the abstract concept of a group, as well. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

Numbers

Integers

The integers Z under addition form a group (described above). In addition to merely being a group, this group is also abelian because

a + b = b + a (commutativity of addition). In Mathematics, commutativity is the ability to change the order of something without changing the end result

The integers are the basic building block for abelian groups, for example every torsion-free group contains (Z, +) as a subgroup. In Abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be ^[41]

The integers, with the operation of multiplication instead of addition, denoted (Z, •) do not form a group. It satisfies the closure, associativity and identity axioms, but fails to have inverses: it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is an integer, but the only solution to the equation ab = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of (Z, •) has a (multiplicative) inverse, so (Z, •) is not a group. ^[42]

Rationals

The wished-for existence of a multiplicative inverses suggests considering fractions

. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object

Such fractions (with integers a and b and b nonzero) are called rational numbers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The set of all such fractions is commonly denoted Q. There is still a minor obstacle for (Q, •), the rationals with multiplication, being a group: since the rational number 0 does not have a multiplicative inverse, (Q, •) is still not a group.

However, the set of all nonzero rational numbers Q \ {0} does form an abelian group under multiplication, denoted (Q \ {0}, •). ^[43] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

The rational numbers (including 0) also form a group under addition. Taking addition and multiplication operations together yields more complicated structures called rings and – if division is possible, such as in Q – fields, which occupy a central position in abstract algebra. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Group theoretic arguments therefore underlie parts of the theory of those entities. ^[44]

Cyclic multiplicative groups

In (Q \ {0}, •), there are the cyclic subgroups

G = {aⁿ, n ∈ Z} ⊂ Q

where aⁿ is the n-th exponentiations of the primitive element a of that group. ^[45] For example, if a is 2 then

G = {. . . , 2⁻², 2⁻¹, 2⁰, 2¹, 2², . . . } = {. . . , 0. 25, 0. 5, 1, 2, 4, . . . }.

This group is an example of a free abelian group of rank one: the rank is one, because G is generated by one element (a or equivalently a⁻¹) and the freeness refers to the fact that no relations between the powers of this generator occur. In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in In Mathematics, the rank, or torsion-free rank, of an Abelian group measures how large a group is in terms of how large a Vector space over the In Mathematics, one method of defining a group is by a presentation. Therefore, G, is isomorphic to the (additive) group of integers (Z, +) above. This example shows that distinguishing between additive and multiplicative groups is merely a matter of notation – group theory treats groups from a purely abstract point of view, forgetting about the concrete nature of the group elements and the group operation.

Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Modular arithmetic the set of Congruence classes Relatively prime to the modulus n form a group under multiplication called the multiplicative ^[46] Its elements are integers not divisible by p, considered modulo p. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers This means that two numbers are considered equivalent if they give the same remainder when divided by p. For example, if p=5, then 4·3=2 in this group, because the usual product 12 is equivalent to 2, for 12 gives rest 2 when divided by 5. The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication. The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that

a · b ≡ 1 (mod p), i. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure e. p divides the difference a · b − 1.

The inverse b can be found by using that the greatest common divisor gcd(a, p) equals 1. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero Hence all group axioms are fulfilled. Actually, this example is similar to (Q\{0}, •) above, because it turns out to be the multiplicative group of nonzero elements in F_p.

Finite groups

Main article: Finite group

A group is called finite if it has finitely many elements. In Mathematics, a finite group is a group which has finitely many elements In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. The number of elements is called order of the group G. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is ^[47] An important class are symmetric groups S_N, the groups of permutations of N letters. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In several fields of Mathematics the term permutation is used with different but closely related meanings For example, the symmetric group on 3 letters S₃ is the group consisting of all possible swaps of the three letters ABC, i. The smallest Non-abelian group has 6 elements It is a Dihedral group with notation D 3 or D 6 e. contains the elements ABC, ACB, . . . , up to CBA, in total 6 (or 3 factorial) elements. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S_N for a suitable N (Cayley's theorem). In Group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup Parallel to the group of symmetries of the square above, S₃ can also be interpreted as the group of symmetries of an equilateral triangle. Properties The area of an equilateral triangle with sides of length a\\!

The order of an element a in a group G is the least positive integer n such that aⁿ = e, where aⁿ represents , i. e. application of the operation • to n copies of the value a. (If • represents multiplication, then aⁿ corresponds to the n^th power of a. ) If no such n exists, then the order of a is said to be infinity. The order of an element is the same as the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any (necessarily finite) subgroup H divides the order of G. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without The Sylow theorems give a partial converse, by asserting the existence of subgroups whose order is any prime power dividing the order of the group, the p-subgroups. In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p

The dihedral group (discussed above) is a finite group of order 8. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections The order of r₁ is 4, because rotating four times by 90° does not change anything. The order of the reflection elements f_v etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem.

Given the notion of finite groups, the obvious aim arises to classify (or list) them. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Z/pZ. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Groups of order p² can also be shown to be abelian. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics. The following list in Mathematics contains the Finite groups of small order Up to Group isomorphism. An intermediate step is the classification of finite simple groups. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. ^[48] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In Mathematics, a trivial group is a group consisting of a single element ^[49] The Jordan-Hölder theorem exhibits simple groups as the building blocks for all finite groups. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces ^[50] Listing all finite simple groups was a major achievement in contemporary group theory. In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families Filling the gaps in the 1982 proof and simplifying it are areas of active research. ^[51] The monstrous moonshine conjectures, proven by 1998 Fields medal winner Richard Borcherds, provide a surprising and deep connection between the largest finite simple sporadic group, called the monster group, and modular functions and string theory. In Mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical Richard Ewen Borcherds (born November 29, 1959) is a British Mathematician specializing in lattices, Number theory, In the Mathematical field of Group theory, a sporadic group is one of the 26 exceptional groups in the Classification of finite simple groups In the Mathematical field of Group theory, the Monster group M or F 1 (also known as the Fischer-Griess Monster or the Friendly Giant In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings ^[52] From the point of view of applications, using modular arithmetic, finite groups are crucial to public-key cryptography. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key ^[53]

Symmetry groups

A periodic wallpaper gives rise to a wallpaper group. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the

Main article: Symmetry group

Symmetry groups are groups consisting of symmetries of given mathematical objects – be they of geometric nature, such as introductory symmetry group of the square, or of algebraic nature. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or ^[54] Conceptually, group theory can be thought of as the study of symmetry. ^[55] Symmetries greatly simplify the study of geometrical or analytical objects. Symmetry in Mathematics occurs not only in Geometry, but also in other branches of mathematics Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Analysis (from Greek ἀνάλυσις, "a breaking up" is the process of breaking a complex topic or substance into smaller parts to gain a This remark is formalized and exploited using the notion of group actions. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. For example, using topological methods such as the monodromy action on the vector space of solutions of certain differential equations, differential Galois theory is able to give group-theoretic criteria when solutions of the equation in question are well-behaved. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, monodromy is the study of how objects from Mathematical analysis, Algebraic topology and algebraic and Differential geometry In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, the Antiderivatives of certain Elementary functions cannot themselves be expressed as elementary functions ^[56] Group actions are also much used in geometric invariant theory. In Mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in Algebraic geometry, used to construct moduli ^[57] Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, or in CD players. In the Mathematical field of Group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple Coding theory is one of the most important and direct applications of Information theory. In Telecommunication and Information theory, forward error correction (FEC is a System of Error control for Data transmission, whereby A Compact Disc player (often written as compact disc player) or CD player, is an electronic device which plays audio Compact Discs CD players are often ^[58]

Lie groups

The circle of center 0 and radius 1 in the complex plane is a Lie group under complex multiplication. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Being a manifold means that every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom). In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

Main article: Lie group

In many situations groups are endowed with an additional structure. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. The most famous examples are Lie groups (in honor of Sophus Lie). Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. They are groups which also have a (compatible) manifold structure, i. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be e. spaces looking locally like some euclidian space of the appropriate dimension. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it ^[59] Because of the manifold structure it is possible to consider continuous paths in the group. In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to For this reason they are also referred to as continuous groups.

Various Lie groups are important tools in physics. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Translations is a three-act play by Irish Playwright Brian Friel written in 1980. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements They can, for instance, be used to construct simple models – imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Lie groups are also of more fundamental importance: Noether's theorem links continuous symmetries to conserved quantities. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves The Poincaré group plays a pivotal role in special relativity and, by implication, for quantum field theories. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In quantum field theory (QFT the forces between particles are mediated by other particles ^[60] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. In quantum field theory (QFT the forces between particles are mediated by other particles Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations ^[61]

General linear group and matrix groups

Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2.

Most of Lie groups important in physics may be described as groups of matrices together with matrix multiplication: the general linear group GL(n, K) consists of all invertible n-by-n matrices with entries in a fixed field K, for example the real or complex numbers. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted ^[62][63]

The subgroups of GL(n, K) are referred to as matrix groups. In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data ^[64] In chemical fields, such as crystallography, space groups and their character tables are used to describe molecular symmetries. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure This article refers to the use of the term character theory in mathematics for the media studies definition see Character theory (Media. Molecular symmetry in Chemistry describes the Symmetry present in Molecules and the classification of molecules according to their symmetry ^[65]

Representation theory

Main article: Representation theory

Representation theory is both an application of the group concept and a major branch of group theory itself. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of ^[66][67] It studies the group by its actions on other spaces. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. A broad class of group representations are linear representations, i. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of e. the group is acting on a vector space, which can be thought of as generalizations of Euclidian space R³. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A representation of G on an n-dimensional complex vector space is simply a group homomorphism

ρ: G → GL(n, C)

from the group to the general linear group. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This way, the group operation, which may be abstractly given, translates to the multiplication of matrices which is accessible to explicit computations.

Given a group action, this gives further means to study the object being acted on. ^[68] On the other hand, it also yields information about the group. Group representations are particularly useful for finite groups, Lie groups, algebraic groups and (locally) compact groups. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact.

Galois groups

Main article: Galois group

Galois groups are groups of substitutions of the solutions of polynomial equations. In Mathematics, a Galois group is a group associated with a certain type of Field extension. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with ^[69][70] For example, the solutions of the quadratic equation ax² + bx + c = 0 are given by

Exchanging "+" and "−" in the expression, i. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. e. permuting the two solutions of the equation can be viewed as a (very easy) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a quintic equation is a Polynomial Equation of degree five Abstract properties of Galois groups (in particular their solvability) associated to polynomials give a criterion which polynomials do have all their solutions expressible by radicals, i. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally e. solutions expressible using solely addition, multiplication, and roots similar to the formula above. In Mathematics, an n th root of a Number a is a number b such that bn = a. ^[71]

See also

Notes

References

General references

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 , Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic Prentice Hall is a leading educational publisher It is an Imprint of Pearson Education Inc
• Devlin, Keith (2000), The Language of Mathematics: Making the Invisible Visible, Owl Books, ISBN 978-0-8050-7254-9 , Chapter 5 provides a layman-accessible explanation of groups. Keith J Devlin is an English Mathematician and Writer. He currently is Executive Director of Stanford University 's Center for the Study
• Dummit, David S. & Foote, Richard M. (2004), Abstract algebra (3rd ed. ), New York: Wiley, MR2286236, ISBN 978-0-471-43334-7 . Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many
• Fulton, William & Harris, Joe (1991), Representation Theory, A First Course, vol. William Fulton (born 1939 is an American algebraic geometer. He received his undergraduate degree from Brown University in 1961 and his Doctorate Joe Harris may be Joe Harris (cricketer Joe Harris (mathematician (born 1951 mathematician Joe Harris (baseball 129, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97495-8 
• Hall, G. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many G. (1967), Applied group theory, American Elsevier Publishing Co. , Inc. , New York, MR0219593 , an elementary introduction
• Herstein, Israel Nathan (1996), Abstract algebra (3rd ed. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Israel Nathan Herstein ( March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was ), Upper Saddle River, NJ: Prentice Hall Inc. , MR1375019, ISBN 978-0-13-374562-7 . Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many
• Herstein, Israel Nathan (1975), Topics in algebra (2nd ed. ), Lexington, Mass. : Xerox College Publishing, MR0356988 
• Lang, Serge (2002), Algebra, vol. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. 211, Graduate Texts in Mathematics, Berlin, New York, MR1878556, ISBN 978-0-387-95385-4 
• Lang, Serge (2005), Undergraduate Algebra (3rd ed. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many ), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3 
• Ledermann, Walter (1953), Introduction to the theory of finite groups, Oliver and Boyd, Edinburgh and London, MR0054593 
• Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6 

Special references

• Artin, Emil (1998), Galois Theory, New York: Dover Publications, ISBN 978-0-486-62342-9 
• Aschbacher, Michael (2004), “The Status of the Classification of the Finite Simple Groups”, Notices of the American Mathematical Society 51 (7): 736–740, ISSN 0002-9920, <http://www.ams.org/notices/200407/fea-aschbacher.pdf> . Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche Notices of the American Mathematical Society is a membership journal of the American Mathematical Society. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication.
• Borel, Armand (1991), Linear algebraic groups, vol. Armand Borel ( 21 May 1923 &ndash 11 August 2003) was a Swiss Mathematician, born in La Chaux-de-Fonds, and was 126 (2nd ed. ), Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 
• Carter, Roger W. (1989), Simple groups of Lie type, New York: John Wiley & Sons, ISBN 978-0-471-50683-6 
• Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Roger W Carter is an Emeritus professor at University of Warwick. John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups & Thurston, William P. (2001), “On three-dimensional space groups”, Beiträge zur Algebra und Geometrie 42 (2): 475–507, MR1865535, ISSN 0138-4821, <http://arxiv.org/abs/math.MG/9911185> 
• Eisenbud, David (1995), Commutative algebra, vol. William Paul Thurston (born October 30, 1946) is an American Mathematician. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. David Eisenbud (born 8 April, 1947) is an American Mathematician. 150, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8 
• Fröhlich, Albrecht (1968), Formal groups, vol. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Albrecht Fröhlich ( 22 May 1916, Munich – 8 November 2001, Cambridge) was a Mathematician famous for his major 74, Lecture notes in mathematics, Berlin, New York: Springer-Verlag 
• Kassel, Christian (1994), Quantum Groups, Springer, ISBN 978-0387943701 
• Kurzweil, Hans & Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, MR2014408, ISBN 978-0-387-40510-0 
• Michler, Gerhard (2006), Theory of finite simple groups, Cambridge University Press, ISBN 978-0-521-86625-5 
• Mumford, David; Fogarty, J. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into & Kirwan, F. (1994), Geometric invariant theory, vol. 34 (3rd ed. ), Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Berlin, New York: Springer-Verlag, MR1304906, ISBN 978-3-540-56963-3 
• Ronan, Mark (2007), Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, Oxford University Press, ISBN 978-0-19-280723-6 
• Serre, Jean-Pierre (1977), Linear representations of finite groups, Berlin, New York: Springer-Verlag, MR0450380, ISBN 978-0-387-90190-9 
• Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8 

Historical references

• Historically important publications in group theory. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. The Princeton University Press is an independent publisher with close connections to Princeton University. Algebra Theory of equations Hisab
• Kleiner, Israel (1986), “The evolution of group theory: a brief survey”, Mathematics Magazine 59 (4): 195–215, MR863090, ISSN 0025-570X, <http://www.jstor.org/sici?sici=0025-570X(198610)59%3A4%3C195%3ATEOGTA%3E2.0.CO%3B2-9> 
• Smith, David Eugene (1906), History of Modern Mathematics, Mathematical Monographs, No. Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. David Eugene Smith PhD LLD (1860&ndash1944 was an American mathematician and educator 1, <http://www.gutenberg.org/etext/8746> 
• Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: Dover Publications, ISBN 978-0-486-45868-7 

External links

• Eric W. Weisstein, Group at MathWorld. Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Group at PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.
• The development of group theory at The MacTutor History of Mathematics archive.