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Encyclopedia > Group (mathematics)
This picture illustrates how the hours on a clock form a group under modular addition.
This picture illustrates how the hours on a clock form a group under modular addition.

In abstract algebra, a group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory. Image File history File links Clock_group. ... Image File history File links Clock_group. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... Group theory is that branch of mathematics concerned with the study of groups. ...


Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular matrices under multiplication and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting. The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...


Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Sphere symmetry group o. ... Sphere symmetry group o. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... For other uses, see Chemistry (disambiguation). ... For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ...

Contents

History

Main article: Group theory#History

Group theory is that branch of mathematics concerned with the study of groups. ...

Explanation for Non-Mathematicians

In mathematics, groups are used in order to abstract away from calculating with concrete numbers (that is, in order to work with symbols instead of numbers). Correspondingly, a group consists of a set of abstract objects or symbols and of an "instruction for calculations" (an operation) that indicates how these objects are to be manipulated. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...


More precisely:


One speaks of a group whenever the following requirements are fulfilled of a set together with an operation that always combines two elements of this set, for example, a x b: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

  1. The combination of two elements of the set yields an element of the same set (|Closure)
  2. The bracketing is unimportant (Associativity): a × (b × c) = (a × b) × c
  3. There is an element that does not cause anything to happen (Identity Element): a × 1 = 1 × a = a
  4. Each element a has a "mirror image" (Inverse element) 1/a that has the property to yield the identity element when combined with a:

a × 1/a = 1/a × a = 1 In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... 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 set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...


Special case: If, in addition, one is also allowed to change the operands, that is if a × b = b × a holds (Commutativity), then we speak of an abelian (or commutative) group. A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...


Examples of groups (all of these are also Abelian groups):

  • the integers Z with the addition operation "+" as binary operation and zero as identity element
  • the rational numbers Bbb Q without zero with multiplication "x" as binary operation and the number one as identity element. Zero has to be excluded because it does not have an inverse element. ("1/0" is undefined.)

The definition of groups is very general. This allows to regard as groups not only sets of numbers with corresponding operations but also other abstract objects and symbols that fulfill the required properties, for example, polygons with their rotations and reflections. (See also: dihedral group) The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Look up polygon in Wiktionary, the free dictionary. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...


Definition

A group (G, *) is a set G with a binary operation * that satisfies the following four axioms: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... For the algebra software named Axiom, see Axiom computer algebra system. ...

Closure: For all a, b in G, the result of a * b is also in G.
Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.
Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.

Some texts omit the explicit requirement of closure, since the closure of the group follows from the definition of a binary operation. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... 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 set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...


Using the identity element property, it can be shown that a group has exactly one identity element. See the proof below. This picture illustrates how the hours on a clock form a group under modular addition. ...


The inverse of an element can also be shown to be unique, and the left- and right-inverses of an element are the same. Some definitions are thus slightly more narrow, substituting the second and third axioms with the concept of a "left (or right) identity element" and a "left (or right) inverse element."


A group (G, *) is often denoted simply G where there is no ambiguity in what the operation is.


Basic concepts in group theory

Order of groups and elements

The order of a group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G. If the order is not finite, then the group is an infinite group, denoted |G| = ∞.


The order of an element a in a group G is the least positive integer n such that an = e, where an is multiplication of a by itself n times (or other suitable composition depending on the group operator). If no such n exists, then the order of a is said to be infinity.


Subgroups

A set H is a subgroup of a group G if it is a subset of G and a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... “Superset” redirects here. ...


If G is a finite group, then so is H. Further, the order of H divides the order of G (Lagrange's Theorem). In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ...


Abelian groups

A group G is said to be an abelian (or commutative) group if the operation is commutative, that is, for all a, b in G, a * b = b * a. A non-abelian group is a group that is not abelian. The term "abelian" is named after the mathematician Niels Abel. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ... Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...


Cyclic groups

A cyclic group is a group whose elements may be generated by successive composition of the operation defining the group being applied to a single element of that group. This single element is called the generator or primitive element of the group. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... A primitive root modulo n is a concept from modular arithmetic in number theory. ...


A multiplicative cyclic group in which G is the group, and a is the generator:


G = { a^n mid n in Z }


An additive cyclic group, with generator a:



If successive composition of the operation defining the group is applied to a non-primitive element of the group, a cyclic subgroup is generated. The order of the cyclic subgroup divides the order of the group. Thus, if the order of a group is prime, all of its elements, except the identity, are primitive elements of the group. In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... 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 set. ...


It is important to note that a group contains all of the cyclic subgroups generated by each of the elements in the group. However, a group constructed from cyclic subgroups is itself not necessarily a cyclic group. For example, a Klein group is not a cyclic group even though it is constructed from two copies of the cyclic group of order 2. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... This article is about the mathematical group. ...


Notation for groups

Groups can use different notation depending on the context and the group operation.

  • Additive groups use + to denote addition, and the minus sign - to denote inverses. For example, a + (-a) = 0 in Z.
  • Multiplicative groups use *, cdot, or the more general 'composition' symbol circ to denote multiplication, and the superscript -1 to denote inverses. For example, a*a-1 = 1. It is very common to drop the * and just write aa-1 instead.
  • Function groups use • to denote function composition, and the superscript -1 to denote inverses. For example, gg-1 = e. It is very common to drop the • and just write gg-1 instead.

Omitting a symbol for an operation is generally acceptable, and leaves it to the reader to know the context and the group operation.


When defining groups, it is standard notation to use parentheses in defining the group and its operation. For example, (H, +) denotes that the set H is a group under addition. For groups like (Zn, +) and (Fn*, *), it is common to drop the parentheses and the operation, e.g. Zn and Fn*. It is also correct to refer to a group by its set identifier, e.g. H or Z, or to define the group in set-builder notation. This article does not cite any references or sources. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...


The identity element e is sometimes known as the "neutral element," and is sometimes denoted by some other symbol, depending on the group:

  • In multiplicative groups, the identity element can be denoted by 1.
  • In invertible matrix groups, the identity element is usually denoted by I.
  • In additive groups, the identity element may be denoted by 0.
  • In function groups, the identity element is usually denoted by f0.

If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products {xs : s in S}; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets (see cosets). In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...


Examples of groups

Some elementary examples of groups in mathematics are given on Group (mathematics). ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

An abelian group: the integers under addition

A familiar group is the group of integers under addition. Let Z be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group. The integers are commonly denoted by the above symbol. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...


Proof:

  • Closure: If a and b are integers then a + b is an integer.
  • Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
  • Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
  • Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.

This group is also abelian because a + b = b + a.


If we extend this example further by considering the integers with both addition and multiplication, which forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group) In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...


An abelian group: the nonzero integers under multiplication modulo P a prime

The nonzero integers under multiplication modulo p a prime form a group. The only non trivial group property to prove is that each element has an inverse. Let a be a nonzero integer not equal to one. Any nonzero integer that p divides equals zero under multiplication mod p. a*a cannot equal a or p will divide a. If a*a equals one, we have found the inverse and we are done. If a*a does not equal one, then a*a*a cannot equal a or a*a or again p will divide a. Continuing in this manner we can construct a*a*a...a up to p-2 times. If we have reached this far, a*a*a...a p-1 times will equal one as there are no more numbers that a*a*a..*a p-1 times can equal.


Cyclic multiplicative groups

In the case of a cyclic multiplicative group G, all of the elements an of the group are generated by the set of all integer exponentiations of a primitive element of that group: In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... “Exponent” redirects here. ...


G = { a^n mid n in Z pmod{m in Z} }


In this example if a is 2 and the operation is the mathematical multiplication operator, then G = {..,2 − 2,2 − 1,20,21,22,23,..} = {..,0.25,0.5,1,2,4,8,..}. The modulo m may bind the group into a finite set with a non-fractional set of elements, since the inverse (and x − 2 , etc.) would be within the set. Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...


Not a group: the integers under multiplication

On the other hand, if we consider the integers with the operation of multiplication, denoted by "·", then (Z,·) is not a group. It satisfies most of the axioms, but fails to have inverses: In mathematics, multiplication is an elementary arithmetic operation. ...

  • Closure: If a and b are integers then a · b is an integer.
  • Associativity: If a, b, and c are integers, then (a · b) · c = a · (b · c).
  • Identity element: 1 is an integer and for any integer a, 1 · a = a · 1 = a.
  • However, 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. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails)

Since not every element of (Z,·) has an inverse, (Z,·) is not a group. It is, however, a commutative monoid, which is a similar structure to a group but does not require inverse elements. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...


An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, the set of all fractions of integers a/b, where a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... For other senses of this word, see zero or 0. ...


However, if we instead use the set of all nonzero rational numbers Q {0}, then (Q {0},·) does form an abelian group.

  • Closure, Associativity, and Identity element axioms are easy to check and follow because of the properties of integers.
  • Inverse elements: The inverse of a/b is b/a and it satisfies the axiom.

We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, the rational numbers form the algebraic structure of a field, allowing the operations of addition, subtraction, multiplication and division. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


A finite nonabelian group: permutations of a set

This example is taken from the larger article on the Dihedral group of order 6

For a more concrete example of a group, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block". The smallest non-Abelian group has 6 elements. ...

Cycle diagram for S3. A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that (ba)2=ab and (ba)3=e, as well as the fact that (ab)2=ba and (ab)3=e The other "loops" are roots of unity so that, for example a2=e.
Cycle diagram for S3. A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that (ba)2=ab and (ba)3=e, as well as the fact that (ab)2=ba and (ab)3=e The other "loops" are roots of unity so that, for example a2=e.

In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions: Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

  • e : RGB → RGB
  • a : RGB → GRB
  • b : RGB → RBG
  • ab : RGB → BRG
  • ba : RGB → GBR
  • aba : RGB → BGR

Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,

  • bb = e,
  • (aba)(aba) = e, and
  • (ab)(ba) = (ba)(ab) = e;

so each of the above actions has an inverse.


By inspection, we can also determine associativity and closure; note for example that

  • (ab)a = a(ba) = aba, and
  • (ba)b = b(ab) = bab.

This group is called the symmetric group on 3 letters, or S3. It has order 6 (or 3 factorial), and is non-abelian (since, for example, abba). Since S3 is built up from the basic actions a and b, we say that the set {a,b} generates it. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... For factorial rings in mathematics, see unique factorisation domain. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...


More generally, we can define a symmetric group from all the permutations of N objects. This group is denoted by SN and has order N factorial. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...


One of the reasons that permutation groups are important is that every finite group can be expressed as a subgroup of a symmetric group SN; this result is Cayley's theorem. In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a...


Simple theorems

Proof: Suppose both e and f are identity elements. Then, by the definition of identity, fe = ef = e and also ef = fe = f. But then e = f.
Therefore the identity element is unique.
  • Every element has exactly one inverse.
Proof: Suppose both b and c are inverses of x. Then, by the definition of an inverse, xb = bx = e and xc = cx = e. But then:
xb = e = xc
xb = xc
bxb = bxc (multiplying on the left by b)
eb = ec (using bx = e)
b = c (neutral element axiom)
Therefore the inverse is unique.

The first two properties actually follow from associative binary operations defined on a set. Given a binary operation on a set, there is at most one identity and at most one inverse for any element. In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

  • You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b.
  • The expression "a1 * a2 * ··· * an" is unambiguous, because the result will be the same no matter where we place parentheses.
  • (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a * b)−1 = b−1 * a−1.
Proof: We will demonstrate that (ab)(b-1a-1) = (b-1a-1)(ab) = e, as required by the definition of an inverse.
(ab)(b − 1a − 1) = a(bb − 1)a − 1 (associativity)
= aea − 1 (definition of inverse)
= aa − 1 (definition of neutral element)
= e (definition of inverse)
And similarly for the other direction.

These and other basic facts that hold for all individual groups form the field of elementary group theory. 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 the same (or equivalent). ... In mathematics, a group (G,*) is usually defined as: G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): A1. ...


Constructing new groups from given ones

Some possible ways to construct new groups from a set of given groups:

  • Subgroups: A subgroup H of a group G is a group.
  • Quotient group: Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.
  • Direct product: If (G,*) and (H,•) are groups, then the set G×H together with the operation (g1,h1)(g2,h2) = (g1*g2,h1h2) is a group. The direct product can also be defined with any number of terms, finite or infinite, by using the Cartesian product and defining the operation coordinate-wise.
  • Semidirect product: If N and H are groups and φ : H → Aut(N) is a group homomorphism, then the semidirect product of N and H with respect to φ is the group (N × H, *), with * defined as
    (n1, h1) * (n2, h2) = (n1 φ(h1) (n2), h1 h2)
  • Direct external sum: The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non-identity coordinates. If the family is finite the direct sum and the product are equivalent.

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, the Cartesian product is a direct product of sets. ... In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. ... 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) · h(v) From this property, one can deduce that h maps the identity element... In group theory, a group G is called the direct sum of a set of subgroups {Hi} if each Hi is a normal subgroup of G each distinct pair of subgroups has trivial intersection, and G = <{Hi}>; in other words, G is generated by the subgroups {Hi}. If G is...

Proving that a set is a group

There are two main methods in proving that a set is a group:

  • Prove that the set is a subgroup of a group;
  • Prove that the set is a group using the definition.

The first method is generally referred to as the "subgroup test" and requires that you prove the following if trying to prove that H is a subgroup: In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... 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 any element in the subset multiplied with any other element in the subset is also in the subset. ...

  • The set H is a non-empty subset of G (i.e. has the identity element inside)
  • H is closed under the same operation as G. (ab is in H and a-1 is in H for all a,b in H)

The second method requires that you prove all the axioms and assumptions in the definition for a set G: The empty set is the set containing no elements. ...

  • G is non-empty;
  • G is closed under the binary operation;
  • G is associative;
  • e is in G (usually follows from non-emptiness);
  • G consists of units.

For finite groups, one only needs to prove that a subset is non-empty and is closed under the ambient group's operation. The word unit means any of several things: One, the first natural number. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...


Generalizations

In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...

  • If we eliminate the requirement that every element have an inverse, then we get a monoid.
  • If we additionally do not require an identity either, then we get a semigroup.
  • Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop.
  • If we additionally do not require an identity, then we get a quasigroup.
  • If we don't require any axioms of the binary operation at all, then we get a magma.

Groupoids, which are similar to groups except that the composition a * b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures. They are special sorts of categories. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...


Supergroups and Hopf algebras are other generalizations. The concept of supergroup is a generalization of a that of group. ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...


Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...


Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond. In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


Formal group laws are certain formal power series which have properties much like a group operation. In mathematics, a formal group law is (roughly speaking) the formal power series analogue of a Lie group. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...


References

  • Herstein, I. N. (1996), Abstract algebra (3 ed.), Upper Saddle River, NJ: Prentice Hall Inc., MR1375019, ISBN 978-0-13-374562-7
  • Dummit, David S. & Richard M. Foote (2004), Abstract algebra (3 ed.), New York: Wiley, MR2286236, ISBN 978-0-471-43334-7
  • Lang, Serge (2002), Algebra, vol. 211, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4

Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ...

See also

Wikibooks
Wikibooks has a book on the topic of
Abstract algebra/Groups

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Please refer to group theory for a general description of the topic. ... In mathematics, a group (G,*) is usually defined as: G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): A1. ... This is a list of group theory topics, by Wikipedia page. ... This is a list of important publications in mathematics, organized by field. ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 – April 10, 1813; b. ... Galois at the age of fifteen from the pencil of a classmate. ...

External links


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