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Encyclopedia > Examples of groups

Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

Contents


Permutations of a set of three elements

Main article: Dihedral group of order 6

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the operation "swap the first block and the second block", and b be the operation "swap the second block and the third block". The smallest non-Abelian group has 6 elements. ...


We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:

Cycle graph for S3 (or D6). 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 ba2=ab and ba3=e, as well as the fact that ab2=ba and ab3=e The other "loops" are roots of unity so that, for example a2=e.
Cycle graph for S3 (or D6). 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 ba2=ab and ba3=e, as well as the fact that ab2=ba and ab3=e The other "loops" are roots of unity so that, for example a2=e.
  • e : RGB → RGB
  • a : RGB → GRB
  • b : RGB → RBG
  • ab : RGB → BRG
  • ba : RGB → GBR
  • aba : RGB → BGR

Note that aa has the effect RGB → GRB → RGB; so we can write aa = e. Similarly, bb = (aba)(aba) = e; (ab)(ba) = (ba)(ab) = e; so every element has an inverse. I, the creator of this image, hereby release it into the public domain. ... In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ...


By inspection, we can determine associativity and closure; note in particular that (ba)b = aba = b(ab).


Since it is built up from the basic operations a and b, we say that the set {a,b} generates this group. The group, called the symmetric group S3, has order 6, and is non-abelian (since, for example, abba). 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, 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. ...


The group of translations of the plane

A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation a o b as follows: first follow the prescription of b, then that of a. For instance, if

a = "move North-East for 2 miles"

and

b = "move South-East for 2 miles"

then

a o b = "move East for sqrt(8) miles"

(see Pythagorean theorem for why this is so, geometrically). The Pythagorean theorem: The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. ...


The set of all translations of the plane with composition as operation forms a group:

  1. If a and b are translations, then a o b is also a translation.
  2. Composition of translations is associative: (a o b) o c = a o (b o c).
  3. The identity element for this group is the translation with prescription "move zero miles in whatever direction you like".
  4. The inverse of a translation is given by walking in the opposite direction for the same distance.

This is an Abelian group and our first (nondiscrete) example of a Lie group: a group whose underlying set is a manifold. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... This page is about a higher mathematics topic. ...


The symmetry group of a square (dihedral group D4)

Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a square concrete slab of a certain thickness. In order to describe its symmetry, we form the set of all those rigid movements of the slab that don't make a visible difference. For instance, if you turn it by 90 degrees clockwise, then it still looks the same, so this movement is one element of our set, let's call it R. We could also flip the slab horizontally so that its underside become up. Again, after performing this movement, the slab looks the same, so this is also an element of our set and we call it T. Then there's of course the movement that does nothing; it's denoted by I. The symmetry group of an object (e. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...


Now if you have two such movements a and b, you can define the composition a o b as above: you first perform the movement b and then the movement a. The result will leave the slab looking like before.


The point is that the set of all those movements, with composition as operation, forms a group. This group is the most concise description of the slab's symmetry. Chemists use symmetry groups of this type to describe the symmetry of crystals.


Let's investigate our slab symmetry group some more. Right now, we have the elements R, T and I, but we can easily form more: for instance R o R, also written as R2, is a 180 degree turn (clockwise or counter-clockwise doesn't matter). R3 is a 270 degree clockwise rotation, or, what is the same thing, a 90 degree counter-clockwise rotation. We also see that T2 = I and also R4 = I. Here's an interesting one: what does R o T do? First flip horizontally, then rotate. Try to visualize that R o T = T o R3. Also, R2 o T is a vertical flip and is equal to T o R2.


This group is actually finite (it has order 8), and we can record everything there is to know about it in a Cayley table (multiplication table): A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ...

Cycle graph for D4. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-a-a2-a3 loop reflects the fact that the successive powers of a are distinct until a4=e. This loop also reflects the fact that successive powers of a3 are distinct until (a3)4=e. The other "loops" are roots of the identity so that, for example b2=e. In the text of the article, R=a, T=b and I=e.
Cycle graph for D4. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-a-a2-a3 loop reflects the fact that the successive powers of a are distinct until a4=e. This loop also reflects the fact that successive powers of a3 are distinct until (a3)4=e. The other "loops" are roots of the identity so that, for example b2=e. In the text of the article, R=a, T=b and I=e.
o I T R R2 R3 RT R2T R3T
I I T R R2 R3 RT R2T R3T
T T I R3T R2T RT R3 R2 R
R R RT R2 R3 I R2T R3T T
R2 R2 R2T R3 I R R3T T RT
R3 R3 R3T I R R2 T RT R2T
RT RT R T R3T R2T I R3 R2
R2T R2T R2 RT T R3T R I R3
R3T R3T R3 R2T RT T R2 R I

For any two elements in the group, the table records what their composition is. Note how every element appears in every row and every column exactly once; this is not a coincidence. You may want to verify some entries. Here we wrote "R3T" as a short hand for R3 o T. Cycle diagram of the D8 group File links The following pages link to this file: Examples of groups Cycle graph (group) Categories: User-created public domain images ... In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ...


Mathematicians know this group as the dihedral group of order 8, and call it either D4 or D8, depending on what textbook they learned from in graduate school. In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ...


This was an example of a non-abelian group: the operation o here is not commutative, which you can see from the table; the table is not symmetrical about the main diagonal. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Matrix groups

If n is some positive integer, we can consider the set of all invertible n by n matrices over the reals, say. This is a group with matrix multiplication as operation. It is called the general linear group, GL(n). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin). For the square matrix section, see square matrix. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ...


If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group, SL(n). Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space. In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...


If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group O(n). Geometrically, this consists of all combinations of rotations and reflections that fix the origin. These are precisely the transformations which preserve lengths and angles. In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...


Finally, if we impose both restrictions, then we get the special orthogonal group SO(n), which consists of rotations only. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...


These groups are our first examples of infinite non-abelian groups. They are also Lie groups. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...


If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n). In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ...


Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups. 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. ...


For example, we have the general linear groups over finite fields. The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121) For a discussion of one of the smallest such groups, GL(2,3), see Visualizing GL(2,p). In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...


Free group on two generators

The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a-1, b and b-1 such that no a appears directly next to an a-1 and no b appears directly next to an b-1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: "abab-1a-1" concatenated with "abab-1a" yields "abab-1a-1abab-1a", which gets reduced to "abaab-1a". One can check that the set of those strings with this operation forms a group with neutral element the empty string ε := "". (Usually the quotation marks are left off, which is why you need the symbol ε!) The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... Generally, string is a thin piece of fiber which is used to tie, bind, or hang other objects. ...


This is another infinite non-abelian group.


Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach-Tarski paradox. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... The Banach-Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...


The set of maps

the sets of maps from a set to a group

Let G be a group and S a nonempty set. The set of maps M(S,G) is itself a group; namely for two maps f,g of S into G we define fg to be the map such that (fg)(x)=f(x)g(x) for every xS and f -1 to be the map such that f -1(x)=f(x)-1.


Take maps f, g, and h in M(S,G). For every x in S, f(x) and g(x) are both in G, and so is (fg)(x). Therefore fg is also in M(S,G), or M(S,G) is closed. For ((fg)h)(x)=(fg)(x)h(x)=(f(x)g(x))h(x)=f(x)(g(x)h(x))=f(x)(gh)(x)=(f(gh))(x), M(S,G) is associative. And there is a map i such that i(x)=e where e is the unit element of G. The map i makes all the functions f in M(S,G) such that if=fi=f, or i is the unit element of M(S,G). Thus, M(S,G) is actually a group.


If G is commutative, (fg)(x)=f(x)g(x)=g(x)f(x)=(gf)(x). Therefore so is M(S,G).


The groups of permutations

Let S be a nonempty set. Let G be the set of bijective mappings of S onto itself. Then G, also denoted by Perm(S), is a group with ordinary composition of mappings. The unit element of G is the identity map of S. For two maps f and g in G are bijective, fg is also bijective. Therefore G is closed. The composition of maps is associative, so is G. Thus, G is a group.


S may be either finite, or infinite.


Some more finite groups

See the list of small groups for some more examples. The following list in mathematics contains the finite groups of small order up to group isomorphism. ...


  Results from FactBites:
 
PlanetMath: examples of groups (1002 words)
This is the automorphism group of the given object and captures its “internal symmetries”;.
In Galois theory, the symmetry groups of field extensions (or equivalently: the symmetry groups of solutions to polynomial equations) are the central object of study; they are called Galois groups.
This is version 26 of examples of groups, born on 2002-06-28, modified 2006-02-25.
CMBI - LHASA: Perception and Functional Groups (532 words)
For each of these groups, a pattern has been defined to describe the group; anything not matching one of the patterns is not recognized as a functional group.
Examples of groups presently unrecognized by LHASA are thioketone, acylsilane, ketene, and sulfonamide.
Secondly, functional groups are used to assess the effects of reaction conditions on portions of the molecule that are not supposed to react (vide infra).
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