applications of symmetry in mathematics, physics & chemistry

8/12/2019 Applications of Symmetry in Mathematics, Physics & Chemistry

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First Edition, 2012

ISBN 978-81-323-4348-6

All rights reserved.

Published by:White Word Public ations

4735/22 Prakashdeep Bldg, Ansari Road, Darya Ganj,Delhi - 110002 Email: [email protected]


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Table of Contents

Chapter 1 -Symmetry

Chapter 2 -Group (Mathematics)

Chapter 3 -Group Action

Chapter 4 -Regular Polytope

Chapter 5 -Lie Point Symmetry


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Chapter 1

Symmetry


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Sphere symmetrical group o.


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Leonardo da Vinci'sVitruvian Man (ca. 1487) is often used as a representation ofsymmetry in the human body and, by extension, the natural universe.

Symmetry (from the Greek: " " = to measure together), 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 perfection. The secondmeaning is a precise and well-defined concept of balance or "patterned self-similarity"that can be demonstrated or proved according to the rules of a formal system: bygeometry, through physics or otherwise.

Although the meanings are distinguishable in some contexts, both meanings of"symmetry" are related and discussed in parallel.


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The "precise" notions of symmetry have various measures and operational definitions.For example, symmetry may be observed:

with respect to the passage of time; as a spatial relationship;

through geometric transformations such as scaling, reflection, and rotation; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even

knowledge itself.

Here we, describes these notions of symmetry from four perspectives. The first is that ofsymmetry in geometry, which is the most familiar type of symmetry for many people.The second perspective is the more general meaning of symmetry in mathematics as awhole. The third perspective describes symmetry as it relates to science and technology.In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses

symmetry in the humanities, covering its rich and varied use in history, architecture, art,and religion.

The opposite of symmetry is asymmetry.

Symmetry in geometry

The most familiar type of symmetry for many people is geometrical symmetry. Formally,this means symmetry under a sub-group of the Euclidean group of isometries in two orthree dimensional Euclidean space. These isometries consist of reflections, rotations,translations and combinations of these basic operations.

Reflection symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry issymmetry with respect to reflection.

In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane ofsymmetry. An object or figure which is indistinguishable from its transformed image iscalled mirror symmetric.

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is

constructed, any two points lying on the perpendicular at equal distances from the axis ofsymmetry are identical. Another way to think about it is that if the shape were to befolded in half over the axis, the two halves would be identical: the two halves are eachother's mirror image. Thus a square has four axes of symmetry, because there are fourdifferent ways to fold it and have the edges all match. A circle has infinitely many axes osymmetry, for the same reason.


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If the letter T is reflected along a vertical axis, it appears the same. Note that this issometimes called horizontal symmetry, and sometimes vertical symmetry. One can betteruse an unambiguous formulation, e.g. "T has a vertical symmetry axis" or "T has left-right symmetry."

The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry arethe kites and the isosceles trapezoids.

For each line or plane of reflection, the symmetry group is isomorphic with Cs, one of thethree types of order two (involutions), hence algebraically C2. The fundamental domainis a half-plane or half-space.

Bilateria (bilateral animals, including humans) are more or less symmetric with respect tothe sagittal plane.

In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is

equivalent with inversion symmetry; in such contexts in modern physics the term P-symmetry is used for both (P stands for parity).

For more general types of reflection there are corresponding more general types ofreflection symmetry. Examples:

with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc.).

with respect to circle inversion

Rotational symmetry

Rotational symmetry is symmetry with respect to some or all rotations in m-dimensionalEuclidean space. Rotations are direct isometries, i.e., isometries preserving orientation.Therefore a symmetry group of rotational symmetry is a subgroup of E +(m).

Symmetry with respect to all rotations about all points implies translational symmetrywith respect to all translations, and the symmetry group is the whole E +(m). This does notapply for objects because it makes space homogeneous, but it may apply for physicallaws.

For symmetry with respect to rotations about a point we can take that point as origin.

These rotations form the special orthogonal group SO(m), the group of mm orthogonalmatrices with determinant 1. For m=3 this is the rotation group.

In another meaning of the word, the rotation group of an object is the symmetry groupwithin E +(n), the group of direct isometries; in other words, the intersection of the fullsymmetry group and the group of direct isometries. For chiral objects it is the same as thefull symmetry group.


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Conversely, if the rotation is slow and the translation is speedy, the coiling angle willapproach 90.

Three main classes of helical symmetry can be distinguished based on the interplay of theangle of coiling and translation symmetries along the axis:

Infinite helical symmetry. If there are no distinguishing features along the lengthof a helix or helix-like object, the object will have infinite symmetry much likethat of a circle, but with the additional requirement of translation along the longaxis of the object to return it to its original appearance. A helix-like object is onethat has at every point the regular angle of coiling of a helix, but which can alsohave a cross section of indefinitely high complexity, provided only that preciselythe same cross section exists (usually after a rotation) at every point along thelength of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries iffor any small rotation of the object around its central axis there exists a point

nearby (the translation distance) on that axis at which the object will appearexactly as it did before. It is this infinite helical symmetry that gives rise to thecurious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices tomove materials along their length, provided that they are combined with a forcesuch as gravity or friction that allows the materials to resist simply rotating alongwith the drill or auger.

n -fold helical symmetry. If the requirement that every cross section of the helicalobject be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still

repeats itself in a regular fashion along the axis of the helical object.Consequently, objects of this type will exhibit a symmetry after a rotation bysome fixed angle and a translation by some fixed distance, but will not ingeneral be invariant for any rotation angle. If the angle (rotation) at which thesymmetry occurs divides evenly into a full circle (360), the result is the helicalequivalent of a regular polygon. This case is calledn-fold helical symmetry , wheren = 360/. This concept can be further generalized to include cases wherem is amultiple of 360that is, the cycle does eventually repeat, but only after morethan one full rotation of the helical object.

Non-repeating helical symmetry. This is the case in which the angle of rotation required to observe the symmetry is irrational. The angle of rotation neverrepeats exactly no matter how many times the helix is rotated. Such symmetriesare created by using a non-repeating point group in two dimensions. DNA is anexample of this type of non-repeating helical symmetry.


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Non-isometric symmetries

A wider definition of geometric symmetry allows operations from a larger group than theEuclidean group of isometries. Examples of larger geometric symmetry groups are:

The group of similarity transformations, i.e. affine transformations represented bya matrix A that is a scalar times an orthogonal matrix. Thus dilations are added,self-similarity is considered a symmetry.

The group of affine transformations represented by a matrix A with determinant 1or 1, i.e. the transformations which preserve area; this adds e.g. obliquereflection symmetry.

The group of all bijective affine transformations.

The group of Mbius transformations which preserve cross-ratios.

In Felix Klein's Erlangen program, each possible group of symmetries defines a geometryin which objects that are related by a member of the symmetry group are considered to beequivalent. For example, the Euclidean group defines Euclidean geometry, whereas thegroup of Mbius transformations defines projective geometry.

Scale symmetry and fractals

Scale symmetry refers to the idea that if an object is expanded or reduced in size, the newobject has the same properties as the original. Scale symmetry is notable for the fact thatit doesnot exist for most physical systems, a point that was first discerned by Galileo.

Simple examples of the lack of scale symmetry in the physical world include thedifference in the strength and size of the legs of elephants versus mice, and theobservation that if a candle made of soft wax was enlarged to the size of a tall tree, itwould immediately collapse under its own weight.

A more subtle form of scale symmetry is demonstrated by fractals. As conceived byBenot Mandelbrot, fractals are a mathematical concept in which the structure of acomplex form looks similar or even exactly the same no matter what degree ofmagnification is used to examine it. A coast is an example of a naturally occurringfractal, since it retains roughly comparable and similar-appearing complexity at everylevel from the view of a satellite to a microscopic examination of how the water laps up

against individual grains of sand. The branching of trees, which enables children to usesmall twigs as stand-ins for full trees in dioramas, is another example.

This similarity to naturally occurring phenomena provides fractals with an everydayfamiliarity not typically seen with mathematically generated functions. As a consequencethey can produce strikingly beautiful results such as the Mandelbrot set. Intriguingly,fractals have also found a place in CG, or computer-generated movie effects, where their


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ability to create very complex curves with fractal symmetries results in more realisticvirtual worlds.

Symmetry in mathematics

In formal terms, we say that a mathematical object issymmetric with respect to a givenmathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the objectform a group. Two objects are symmetric to each other with respect to a given group ofoperations if one is obtained from the other by some of the operations (and vice versa).

Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X , can be modeledas a group actiong : G X X , where the image ofg inG and x in X is written asg x. If,for someg, g x = y then x and y are said to be symmetrical to each other. For each object

x, operationsg for whichg x = x form a group, thesymmetry group of the object, asubgroup ofG. If the symmetry group of x is the trivial group then x is said to beasymmetric , otherwisesymmetric . A general example is thatG is a group of bijectionsg: V V acting on the set of functions x: V W by (gx)(v) = x(g1(v)) (or a restricted setof such functions that is closed under the group action). Thus a group of bijections ofspace induces a group action on "objects" in it. The symmetry group of x consists of allg for which x(v) = x(g(v)) for allv. G is the symmetry group of the space itself, and of anyobject that is uniform throughout space. Some subgroups ofG may not be the symmetrygroup of any object. For example, if the group contains for everyv andw inV a g suchthatg(v) =w, then only the symmetry groups of constant functions x contain that group.However, the symmetry group of constant functions isG itself.

In a modified version for vector fields, we have (gx)(v) =h(g, x(g1(v))) whereh rotatesany vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors)according to rotation and inversion ing. The symmetry group of x consists of allg forwhich x(v) =h(g, x(g(v))) for allv. In this case the symmetry group of a constant functionmay be a proper subgroup ofG: a constant vector has only rotational symmetry withrespect to rotation about an axis if that axis is in the direction of the vector, and onlyinversion symmetry if it is zero.

For a common notion of symmetry in Euclidean space,G is the Euclidean group E (n), thegroup of isometries, andV is the Euclidean space. Therotation group of an object is the

symmetry group ifG is restricted to E +

(n), the group of direct isometries. Objects can bemodeled as functions x, of which a value may represent a selection of properties such ascolor, density, chemical composition, etc. Depending on the selection we consider justsymmetries of sets of points ( x is just a Boolean function of positionv), or, at the otherextreme, e.g. symmetry of right and left hand with all their structure.

For a given symmetry group, the properties of part of the object, fully define the wholeobject. Considering points equivalent which, due to the symmetry, have the same


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properties, the equivalence classes are the orbits of the group action on the space itself.We need the value of x at one point in every orbit to define the full object. A set of suchrepresentatives forms a fundamental domain. The smallest fundamental domain does nothave a symmetry; in this sense, one can say that symmetry relies upon asymmetry.

An object with a desired symmetry can be produced by choosing for every orbit a singlefunction value. Starting from a given object x we can e.g.:

take the values in a fundamental domain (i.e., add copies of the object)

take for each orbit some kind of average or sum of the values of x at the points ofthe orbit (ditto, where the copies may overlap)

If it is desired to have no more symmetry than that in the symmetry group, then the objecto be copied should be asymmetric.

As pointed out above, some groups of isometries are not the symmetry group of anyobject, except in the modified model for vector fields. For example, this applies in 1D forthe group of all translations. The fundamental domain is only one point, so we can notmake it asymmetric, so any "pattern" invariant under translation is also invariant underreflection (these are the uniform "patterns").

In the vector field version continuous translational symmetry does not imply reflectionalsymmetry: the function value is constant, but if it contains nonzero vectors, there is noreflectional symmetry. If there is also reflectional symmetry, the constant function valuecontains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic

field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. Forvectors (in particular the current density) we have symmetry in every plane perpendiculato the cylinder, as well as cylindrical symmetry. This cylindrical symmetry withoutmirror planes through the axis is also only possible in the vector field version of thesymmetry concept. A similar example is a cylinder rotating about its axis, wheremagnetic field and current density are replaced by angular momentum and velocity,respectively.

A symmetry group is said to act transitively on a repeated feature of an object if, forevery pair of occurrences of the feature there is a symmetry operation mapping the first tothe second. For example, in 1D, the symmetry group of {...,1,2,5,6,9,10,13,14,...} acts

transitively on all these points, while {...,1,2,3,5,6,7,9,10,11,13,14,15,...} doesnot acttransitively on all points. Equivalently, the first set is only one conjugacy class withrespect to isometries, while the second has two classes.


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Symmetric funct ions

A symmetric function is a function which is unchanged by any permutation of itsvariables. For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2

yz is not.

A function may be unchanged by a sub-group of all the permutations of its variables. Forexample,ac + 3ab + bc is unchanged ifa andb are exchanged; its symmetry group isisomorphic to C2.

Symmetry in logic

A dyadic relation R is symmetric if and only if, whenever it's true that Rab , it's true that Rba . Thus, is the same age as is symmetrical, for if Paul is the same age as Mary, thenMary is the same age as Paul.

Symmetric binary logical connectives are "and" ( , , or &), "or" ( ), "biconditional"(if and only if) (), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").

Symmetry in science

Symmetry in physics

Symmetry in physics has been generalized to mean invariancethat is, lack of anyvisible changeunder any kind of transformation, for example arbitrary coordinatetransformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate insymmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in hiswidely read 1972 article More is Different that "it is only slightly overstating the case tosay that physics is the study of symmetry."

Symmetry in physical objects

Classical objects

Although an everyday object may appear exactly the same after a symmetry operationsuch as a rotation or an exchange of two identical parts has been performed on it, it isreadily apparent that such a symmetry is true only as an approximation for any ordinary physical object.

For example, if one rotates a precisely machined aluminum equilateral triangle 120degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is thateach corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as


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optical or electron microscopes will not be fooled; he will immediately recognize that theobject has been rotated by looking for details such as crystals or minor deformities.

Such simple thought experiments show that assertions of symmetry in everyday physicalobjects are always a matter of approximate similarity rather than of precise mathematical

sameness. The most important consequence of this approximate nature of symmetries ineveryday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper symmetries of space and time play a major role in classical physicsthat is, the physics of large, everyday objects.

Quantum objects

Remarkably, there exists a realm of physics for which mathematical assertions of simplesymmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such aselectrons, protons, light, and atoms.

Unlike everyday objects, objects such as electrons have very limited numbers ofconfigurations, called states, in which they can exist. This means that when symmetryoperations such as exchanging the positions of components are applied to them, theresulting new configurations often cannot be distinguished from the originals no matterhow diligent an observer is. Consequently, for sufficiently small and simple objects thegeneric mathematical symmetry assertionF ( x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the realworld.

Consequences of quantum symmetry

While it makes sense that symmetries could become exact when applied to very simpleobjects, the immediate intuition is that such a detail should not affect the physics of suchobjects in any significant way. This is in part because it is very difficult to view theconcept of exact similarity as physically meaningful. Our mental picture of suchsituations is invariably the same one we use for large objects: We picture objects orconfigurations that are very, very similar, but for which if we could "look closer" wewould still be able to tell the difference.

However, the assumption that exact symmetries in very small objects should not makeany difference in their physics was discovered in the early 1900s to be spectacularly

incorrect. The situation was succinctly summarized by Richard Feynman in the directtranscripts of his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles . (Unfortunately, the quote was edited out of the printed version of the samelecture.)

"... if there is a physical situation in which it is impossible to tell which way ithappened, italways interferes; itnever fails."


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The word "interferes" in this context is a quick way of saying that such objects fall underthe rules of quantum mechanics, in which they behave more like waves that interfere thanlike everyday large objects.

In short, when an object becomes so simple that a symmetry assertion of the formF(x) =

x becomes an exact statement of experimentally verifiable sameness, x ceases to followthe rules of classical physics and must instead be modeled using the more complexandoften far less intuitiverules of quantum physics.

This transition also provides an important insight into why the mathematics of symmetryare so deeply intertwined with those of quantum mechanics. When physical systemsmake the transition from symmetries that are approximate to ones that are exact, themathematical expressions of those symmetries cease to be approximations and aretransformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes soclose that it is difficult to separate the two.

Generalizations of symmetry

If we have a given set of objects with some structure, then it is possible for a symmetry tomerely convert only one object into another, instead of acting upon all possible objectssimultaneously. This requires a generalization from the concept of symmetry group tothat of a groupoid. Indeed, A. Connes in his book `Non-commutative geometry' writesthat Heisenberg discovered quantum mechanics by considering the groupoid oftransitions of the hydrogen spectrum.

The notion of groupoid also leads to notions of multiple groupoids, namely sets with

many compatible groupoid structures, a structure which trivialises to abelian groups ifone restricts to groups. This leads to prospects of `higher order symmetry' which have been a little explored, as follows.

The automorphisms of a set, or a set with some structure, form a group, which models ahomotopy 1-type. The automorphisms of a groupG naturally form a crossed module

, and crossed modules give an algebraic model of homotopy 2-types. Atthe next stage, automorphisms of a crossed module fit into a structure known as a crossedsquare, and this structure is known to give an algebraic model of homotopy 3-types. It isnot known how this procedure of generalising symmetry may be continued, althoughcrossedn-cubes have been defined and used in algebraic topology, and these structuresare only slowly being brought into theoretical physics.Physicists have come up with other directions of generalization, such as supersymmetryand quantum groups, yet the different options are indistinguishable during variouscircumstances


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Chapter 2

Group (Mathematics)

The possible manipulations of this Rubik's Cube form a group.

In mathematics, agroup is an algebraic structure consisting of a set together with anoperation that combines any two of its elements to form a third element. To qualify as agroup, the set and the operation must satisfy a few conditions called group axioms,namely closure, associativity, identity and invertibility. Many familiar mathematicalstructures such as number systems obey these axioms: for example, the integers endowedwith the addition operation form a group. However, the abstract formalization of the


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group axioms, detached as it is from the concrete nature of any particular group and itsoperation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.The ubiquity of groups in numerous areas within and outside mathematics makes them acentral organizing principle of contemporary mathematics.

Groups share a fundamental kinship with the notion of symmetry. A symmetry groupencodes symmetry features of a geometrical object: it consists of the set oftransformations that leave the object unchanged, and the operation of combining twosuch transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academicdisciplines. Matrix groups, for example, can be used to understand fundamental physicallaws underlying special relativity and symmetry phenomena in molecular chemistry.

The concept of a group arose from the study of polynomial equations, starting withvariste Galois in the 1830s. After contributions from other fields such as number theory

and geometry, the group notion was generalized and firmly established around 1870.Modern group theorya very active mathematical disciplinestudies groups in theirown right. To explore groups, mathematicians have devised various notions to breakgroups into smaller, better-understandable pieces, such as subgroups, quotient groups andsimple groups. In addition to their abstract properties, group theorists also study thedifferent ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification ofinite simple groups completed in 1983. Since the mid-1980s, geometric group theory,which studies finitely generated groups as geometric objects, has become a particularlyactive area in group theory.

Definition and illustration

First example: the integers

One of the most familiar groups is the set of integersZ which consists of the numbers

...,4, 3, 2, 1, 0, 1, 2, 3, 4, ...

The following properties of integer addition serve as a model for the abstract groupaxioms given in the definition below.

1. For any two integersa andb, the suma + b is also an integer. In other words, the process of adding integers two at a time always yields an integer, not some othertype of number such as a fraction. This property is known asclosure underaddition.

2. For all integersa , b andc, (a + b) +c = a + (b + c). Expressed in words, addinga to b first, and then adding the result toc gives the same final result as addinga tothe sum ofb andc, a property known asassociativity .


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3. If a is any integer, then 0 +a = a + 0 =a . Zero is called theidentity element ofaddition because adding it to any integer returns the same integer.

4. For every integera , there is an integerb such thata + b = b + a = 0. The integerb is called theinverse element of the integera and is denoteda .

The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand thesestructures as a collective, the following abstract definition is developed.

Definition

A group is a set,G, together with an operation (called thegroup law ofG) thatcombines any two elementsa andb to form another element, denoteda b orab . Toqualify as a group, the set and operation, (G, ), must satisfy four requirements known asthegroup axioms :

ClosureFor alla , b inG, the result of the operation,a b, is also inG.Associativity

For alla , b andc inG, (a b) c = a (b c).Identity element

There exists an elemente inG, such that for every elementa inG, the equatione a = a e = a holds. The identity element of a groupG is often written as 1 or 1G,a notation inherited from the multiplicative identity.

Inverse elementFor eacha inG, there exists an elementb inG such thata b = b a = 1G.

The order in which the group operation is carried out can be significant. In other words,the result of combining elementa with elementb need not yield the same result ascombining elementb with elementa ; the equation

a b = b a

may not always be true. This equation does always hold in the group of integers underaddition, becausea + b = b + a for any two integers (commutativity of addition).However, it does not always hold in the symmetry group below. Groups for which theequationa b = b a always holds are calledabelian (in honor of Niels Abel). Thus, theinteger addition group is abelian, but the following symmetry group is not.

The setG is called theunderlying set of the group (G, ). Often the group's underlying setG is used as a short name for the group (G, ). Along the same lines, sometimes ashorthand expression such as "a subset of the groupG" is used when what is actuallymeant is "a subset of the underlying setG of the group (G, )." Usually, it is clear fromthe context whether a symbol likeG refers to a group or to an underlying set.


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Second example: a symmetry group

The symmetries (i.e., rotations and reflections) of a square form a group called a dihedralgroup, and denoted D4. The following symmetries occur:

id (keeping it asis) r 1 (rotation by 90right)

r 2 (rotation by180 right)

r 3 (rotation by 270right)

f v (vertical flip) f h (horizontal flip) f d (diagonal flip)f c (counter-

diagonal flip)

The elements of the symmetry group of the square (D4). The vertices are coloredand numbered only to visualize the operations.

the identity operation leaving everything unchanged, denoted id; rotations of the square by 90 right, 180 right, and 270 right, denoted by

r 1, r 2 and r 3, respectively; reflections about the vertical and horizontal middle line (f h and f v), or

through the two diagonals (f d and f c).

The defining operation of this group is function composition: The eight symmetries arefunctions from the square to the square, and two symmetries are combined by composingthem as functions, that is, applying the first one to the square, and the second one to theresult of the first application. The result of performing firsta and thenb is writtensymbolically from right to left as

b a ("apply the symmetryb after performing the symmetrya"). The right-to-leftnotation is the same notation that is used for composition of functions.

The group table on the right lists the results of all such compositions possible. Forexample, rotating by 270 right (r 3) and then flipping horizontally (f h) is the same as performing a reflection along the diagonal (f d). Using the above symbols, highlighted in blue in the group table:


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f h r 3 = f d.

Group table of D4

id r 1 r 2 r 3 f v f h f d f c

id id r 1 r 2 r 3 f v f h f d f c

r 1 r 1 r 2 r 3 id f c f d f v f h

r 2 r 2 r 3 id r 1 f h f v f c f d

r 3 r 3 id r 1 r 2 f d f c f h f v

f v f v f d f h f c id r 2 r 1 r 3

f h f h f c f v f d r 2 id r 3 r 1

f d f d f h f c f v r 3 r 1 id r 2

f c f c f v f d f h r 1 r 3 r 2 id

The elements id, r 1, r 2, and r 3 form a subgroup,highlighted in red (upper left region). A left andright coset of this subgroup is highlighted ingreen (in the last row) and yellow (last column),respectively.


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Given this set of symmetries and the described operation, the group axioms can beunderstood as follows:

1. The closure axiom demands that the compositionb a of any two symmetriesa andb is also a symmetry. Another example for the group operation is

r 3 f h = f c,

i.e. rotating 270 right after flipping horizontally equals flipping along thecounter-diagonal (f c). Indeed every other combination of two symmetries stillgives a symmetry, as can be checked using the group table.

2. The associativity constraint deals with composing more than two symmetries:Starting with three elementsa , b andc of D4, there are two possible ways of usingthese three symmetries in this order to determine a symmetry of the square. Oneof these ways is to first composea andb into a single symmetry, then to compose

that symmetry withc. The other way is to first composeb andc, then to composethe resulting symmetry witha . The associativity condition

(a b) c = a (b c)

means that these two ways are the same, i.e., a product of many group elementscan be simplified in any order. For example, (f d f v) r 2 = f d (f v r 2) can bechecked using the group table at the right

(f d f v) r 2 = r 3 r 2 = r 1, which equalsf d (f v r 2) = f d f h = r 1.

While associativity is true for the symmetries of the square and addition ofnumbers, it is not true for all operations. For instance, subtraction of numbers isnot associative: (7 3) 2 = 2 is not the same as 7 (3 2) = 6.

3. The identity element is the symmetry id leaving everything unchanged: for anysymmetrya , performing id aftera (ora after id) equalsa , in symbolic form,

id a = a ,a id =a .

4. An inverse element undoes the transformation of some other element. Everysymmetry can be undone: each of transformationsidentity id, the flips f h, f v, f d,f c and the 180 rotation r 2 is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r 3 and r 1 are eachother's inverse, because rotating 90 and then rotation 270 (or vice versa) yields arotation over 360 which leaves the square unchanged. In symbols,

f h f h = id,


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r 3 r 1 = r 1 r 3 = id.

In contrast to the group of integers above, where the order of the operation is irrelevant, idoes matter in D4: f h r 1 = f c but r 1 f h = f d. In other words, D4 is not abelian, whichmakes the group structure more difficult than the integers introduced first.

History

The modern concept of an abstract group developed out of several fields of mathematics.The original motivation for group theory was the quest for solutions of polynomialequations of degree higher than 4. The 19th-century French mathematician varisteGalois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave acriterion for the solvability of a particular polynomial equation in terms of the symmetrygroup of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in

particular by Augustin Louis Cauchy. Arthur Cayley'sOn the theory of groups, asdepending on the symbolic equation n = 1 (1854) gives the first abstract definition of afinite group.

Geometry was a second field in which groups were used systematically, especiallysymmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometriessuch as hyperbolic and projective geometry had emerged, Klein used group theory toorganize them in a more coherent way. Further advancing these ideas, Sophus Liefounded the study of Lie groups in 1884.

The third field contributing to group theory was number theory. Certain abelian group

structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847,Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax bydeveloping groups describing factorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started withCamille Jordan'sTrait des substitutions et des quations algbriques (1870). Walthervon Dyck (1882) gave the first statement of the modern definition of an abstract group.As of the 20th century, groups gained wide recognition by the pioneering work ofFerdinand Georg Frobenius and William Burnside, who worked on representation theoryof finite groups, Richard Brauer's modular representation theory and Issai Schur's papers

The theory of Lie groups, and more generally locally compact groups was pushed byHermann Weyl, lie Cartan and many others. Its algebraic counterpart, the theory ofalgebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by pivotal work of Armand Borel and Jacques Tits.

The University of Chicago's 196061 Group Theory Year brought together grouptheorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying thefoundation of a collaboration that, with input from numerous other mathematicians,


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equation bya 1 gives the solution x = x a a 1 = b a 1. Similarly there is exactly onesolution y inG to the equationa y = b, namely y = a 1 b. In general, x and y need notagree.

A consequence of this is that multiplying by a group elementg is a bijection.

Specifically, ifg is an element of the groupG, there is a bijection fromG to itself calledleft translation by g sendingh G tog h. Similarly,right translation by g is a bijection fromG to itself sendingh toh g. IfG is abelian, left and right translation by agroup element are the same.

Basic conceptsThe following sections use mathematical symbols such as X ={ x , y , z } to denote a set Xcontaining elements x, y, and z, or alternatively x X to restate that x is an element ofX. The notation f : X Y means f is a function assigning to every element of X anelement of Y.

To understand groups beyond the level of mere symbolic manipulations as above, morestructural concepts have to be employed. There is a conceptual principle underlying all ofthe following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to becompatible with the group operation. This compatibility manifests itself in the following notions invarious ways. For example, groups can be related to each other via functions called grouphomomorphisms. By the mentioned principle, they are required to respect the groupstructures in a precise sense. The structure of groups can also be understood by breakingthem into pieces called subgroups and quotient groups. The principle of "preservingstructures"a recurring topic in mathematics throughoutis an instance of working in acategory, in this case the category of groups.

Group homomorphisms

Group homomorphisms are functions that preserve group structure. A functiona: G H between two groups (G,) and ( H ,*) is a homomorphism if the equation

a(g k ) =a(g) *a(k )

holds for all elementsg, k inG. In other words, the result is the same when performingthe group operation after or before applying the mapa . This requirement ensures thata(1G) = 1 H , and alsoa(g)1 = a(g1) for allg inG. Thus a group homomorphism respectsall the structure ofG provided by the group axioms.

Two groupsG and H are called isomorphic if there exist group homomorphismsa : G H andb: H G, such that applying the two functions one after another (in each of thetwo possible orders) equal the identity function ofG and H , respectively. That is,a(b(h))= h andb(a(g)) =g for anyg inG andh in H . From an abstract point of view, isomorphicgroups carry the same information. For example, proving thatg g = 1G for some element


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g ofG is equivalent to proving thata(g) a(g) = 1H, because applyinga to the firstequality yields the second, and applyingb to the second gives back the first.

Subgroups

Informally, asubgroup is a group H contained within a bigger one,G. Concretely, theidentity element ofG is contained in H , and wheneverh1 andh2 are in H , then so areh1 h2 andh11, so the elements of H , equipped with the group operation onG restricted to H ,indeed form a group.

In the example above, the identity and the rotations constitute a subgroup R = {id, r 1, r 2,r 3}, highlighted in red in the group table above: any two rotations composed are still arotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations270 for 90, 180 for 180, and 90 for 270 (note that rotation in the opposite directionis not defined). The subgroup test is a necessary and sufficient condition for a subset H ofa groupG to be a subgroup: it is sufficient to check thatg1h H for all elementsg, h

H . Knowing the subgroups is important in understanding the group as a whole.Given any subsetS of a groupG, the subgroup generated byS consists of products ofelements ofS and their inverses. It is the smallest subgroup ofG containingS . In theintroductory example above, the subgroup generated by r 2 and f v consists of these twoelements, the identity element id and f h = f v r 2. Again, this is a subgroup, becausecombining any two of these four elements or their inverses (which are, in this particularcase, these same elements) yields an element of this subgroup.

Cosets

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a flip is performed,the square never gets back to the r 2 configuration by just applying the rotation operations(and no further flips), i.e. the rotation operations are irrelevant to the question whether aflip has been performed. Cosets are used to formalize this insight: a subgroup H definesleft and right cosets, which can be thought of as translations of H by arbitrary groupelementsg. In symbolic terms, theleft andright coset of H containingg are

gH = {g h , h H } and Hg = {h g , h H }, respectively.

The cosets of any subgroup H form a partition ofG; that is, the union of all left cosets isequal toG and two left cosets are either equal or have an empty intersection. The firstcaseg1 H = g2 H happens precisely wheng11 g2 H , i.e. if the two elements differ byan element of H . Similar considerations apply to the right cosets of H . The left and rightcosets of H may or may not be equal. If they are, i.e. for allg inG, gH = Hg , then H issaid to be anormal subgroup . One may then simply refer to N as the set of cosets.

In D4, the introductory symmetry group, the left cosetsgR of the subgroup R consistingof the rotations are either equal to R, ifg is an element of R itself, or otherwise equal toU


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= f c R = {f c, f v, f d, f h} (highlighted in green). The subgroup R is also normal, because f c R =U = Rf c and similarly for any element other than f c.

Quotient groups

In addition to disregarding the internal structure of a subgroup by considering its cosets,it is desirable to endow this coarser entity with a group law calledquotient group or factor group . For this to be possible, the subgroup has to be normal. Given any normalsubgroup N , the quotient group is defined by

G / N = {gN , g G}, "G modulo N ".

This set inherits a group operation (sometimes called coset multiplication, or cosetaddition) from the original groupG: (gN ) (hN ) = (gh) N for allg andh inG. Thisdefinition is motivated by the idea (itself an instance of general structural considerationsoutlined above) that the mapG G / N that associates to any elementg its cosetgN be a

group homomorphism, or by general abstract considerations called universal properties.The coseteN = N serves as the identity in this group, and the inverse ofgN in the quotientgroup is (gN )1 = (g1) N .

R U

R R U

U U R

Group table of the quotient group D4 / R.

The elements of the quotient group D4 / R are R itself, which represents the identity, andU = f v R. The group operation on the quotient is shown at the right. For example,U U =f v R f v R = (f v f v) R = R. Both the subgroup R = {id, r 1, r 2, r 3}, as well as thecorresponding quotient are abelian, whereas D4 is not abelian. Building bigger groups bysmaller ones, such as D4 from its subgroup R and the quotient D4 / R is abstracted by anotion called semidirect product.

Quotient and subgroups together form a way of describing every group by its presentation : any group is the quotient of the free group over thegenerators of the group,quotiented by the subgroup ofrelations . The dihedral group D4, for example, can begenerated by two elementsr and f (for example,r = r 1, the right rotation and f = f v thevertical (or any other) flip), which means that every symmetry of the square is a finitecomposition of these two symmetries or their inverses. Together with the relations

r 4 = f 2 = (r f )2 = 1,


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the group is completely described. A presentation of a group can also be used to constructhe Cayley graph, a device used to graphically capture discrete groups.

Sub- and quotient groups are related in the following way: a subset H ofG can be seen asan injective map H G, i.e. any element of the target has at most one element that maps

to it. The counterpart to injective maps are surjective maps (every element of the target ismapped onto), such as the canonical mapG G / N . Interpreting subgroup and quotientsin light of these homomorphisms emphasizes the structural concept inherent to thesedefinitions alluded to in the introduction. In general, homomorphisms are neitherinjective nor surjective. Kernel and image of group homomorphisms and the firstisomorphism theorem address this phenomenon.

Examples and applications

A periodic wallpaper pattern gives rise to a wallpaper group.


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The fundamental group of a plane minus a point (bold) consists of loops around themissing point. This group is isomorphic to the integers.

Examples and applications of groups abound. A starting point is the groupZ of integerswith addition as group operation, introduced above. If instead of addition multiplicationis considered, one obtains multiplicative groups. These groups are predecessors ofimportant constructions in abstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects areoften examined by associating groups to them and studying the properties of thecorresponding groups. For example, Henri Poincar founded what is now called algebraitopology by introducing the fundamental group. By means of this connection, topologica properties such as proximity and continuity translate into properties of groups. Forexample, elements of the fundamental group are represented by loops. The second imageat the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. Thefundamental group of the plane with a point deleted turns out to be infinite cyclic,generated by the orange loop (or any other loop winding once around the hole). This waythe fundamental group detects the hole.

In more recent applications, the influence has also been reversed to motivate geometricconstructions by a group-theoretical background. In a similar vein, geometric grouptheory employs geometric concepts, for example in the study of hyperbolic groups.Further branches crucially applying groups include algebraic geometry and numbertheory.

In addition to the above theoretical applications, many practical applications of groupsexist. Cryptography relies on the combination of the abstract group theory approachtogether with algorithmical knowledge obtained in computational group theory, in


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particular when implemented for finite groups. Applications of group theory are notrestricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.

Numbers

Many number systems, such as the integers and the rationals enjoy a naturally givengroup structure. In some cases, such as with the rationals, both addition andmultiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Furtherabstract algebraic concepts such as modules, vector spaces and algebras also form groups

Integers

The group of integersZ under addition, denoted (Z , +), has been described above. Theintegers, with the operation of multiplication instead of addition, (Z , ) donot form a

group. The closure, associativity and identity axioms are satisfied, but inverses do notexist: for example,a = 2 is an integer, but the only solution to the equationa b = 1 inthis case isb = 1/2, which is a rational number, but not an integer. Hence not everyelement ofZ has a (multiplicative) inverse.

Rationals

The desire for the existence of multiplicative inverses suggests considering fractions

Fractions of integers (withb nonzero) are known as rational numbers. The set of all suchfractions is commonly denotedQ . There is still a minor obstacle for (Q , ), the rationalswith multiplication, being a group: because the rational number 0 does not have amultiplicative inverse (i.e., there is no x such that x 0 = 1), (Q , ) is still not a group.

However, the set of allnonzero rational numbersQ \ {0} = {q Q , q 0} does form anabelian group under multiplication, denoted (Q \ {0}, ). Associativity and identityelement axioms follow from the properties of integers. The closure requirement stillholds true after removing zero, because the product of two nonzero rationals is neverzero. Finally, the inverse ofa /b is b/a , therefore the axiom of the inverse element is

satisfied.The rational numbers (including 0) also form a group under addition. Intertwiningaddition and multiplication operations yields more complicated structures called ringsandif division is possible, such as inQ fields, which occupy a central position inabstract algebra. Group theoretic arguments therefore underlie parts of the theory of thosentities.


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Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group ofintegers modulo p. Its elements are integers not divisible by p, considered modulo p, i.e.two numbers are considered equivalent if their difference is divisible by p. For example,

if p = 5, there are exactly four group elements 1, 2, 3, 4: multiples of 5 are excluded and 6and4 are both equivalent to 1 etc. The group operation is given by multiplication.Therefore, 4 4 = 1, because the usual product 16 is equivalent to 1, for 5 divides 16 1= 15, denoted

16 1 (mod 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 undermultiplication. The identity element is 1, as usual for a multiplicative group, and theassociativity follows from the corresponding property of integers. Finally, the inverse

element axiom requires that given an integera not divisible by p, there exists an integerb such that

a b 1 (mod p), i.e. p divides the differencea b 1.

The inverseb can be found by using Bzout's identity and the fact that the greatestcommon divisor gcd(a , p) equals 1. In the case p = 5 above, the inverse of 4 is 4, and theinverse of 3 is 2, as 3 2 = 6 1 (mod 5). Hence all group axioms are fulfilled. Actually,this example is similar to (Q \{0}, ) above, because it turns out to be the multiplicativegroup of nonzero elements in the finite fieldF p, denotedF p. These groups are crucial to public-key cryptography.


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Cyclic groups

The 6th complex roots of unity form a cyclic group. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2.

A cyclic group is a group all of whose elements are powers (when the group operation iswritten additively, the term 'multiple' can be used) of a particular elementa . Inmultiplicative notation, the elements of the group are:

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

wherea2 meansa a , anda 3 stands fora 1 a 1 a 1=(a a a)1 etc Such an elementa is called a generator or a primitive element of the group.

A typical example for this class of groups is the group ofn-th complex roots of unity,given by complex numbers z satisfying zn = 1 (and whose operation is multiplication).Any cyclic group withn elements is isomorphic to this group. Using some field theory,the groupF p can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 =3, 32 = 9 4, 33 2, and 34 1.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero elementa , all the powers ofa are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to (Z , +), the


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group of integers under addition introduced above. As these two prototypes are bothabelian, so is any cyclic group.

The study of abelian groups is quite mature, including the fundamental theorem offinitely generated abelian groups; and reflecting this state of affairs, many group-related

notions, such as center and commutator, describe the extent to which a given group is notabelian.

Symmetry groups

Symmetry groups are groups consisting of symmetries of given mathematical objectsbethey of geometric nature, such as the introductory symmetry group of the square, or ofalgebraic nature, such as polynomial equations and their solutions. Conceptually, grouptheory can be thought of as the study of symmetry. Symmetries in mathematics greatlysimplify the study of geometrical or analytical objects. A group is said to act on anothermathematical object X if every group element performs some operation on X compatibly

to the group law. In the rightmost example below, an element of order 7 of the (2,3,7)triangle group acts on the tiling by permuting the highlighted warped triangles (and theother ones, too). By a group action, the group pattern is connected to the structure of theobject being acted on.


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Rotations and flips form the symmetry group of a great icosahedron.

In chemical fields, such as crystallography, space groups and point groups describemolecular symmetries and crystal symmetries. These symmetries underlie the chemicaland physical behavior of these systems, and group theory enables simplification ofquantum mechanical analysis of these properties. For example, group theory is used toshow that optical transitions between certain quantum levels cannot occur simply becauseof the symmetry of the states involved.

Not only are groups useful to assess the implications of symmetries in molecules, butsurprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particularground state of lower symmetry from a set of possible ground states that are related toeach other by the symmetry operations of the molecule.


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Likewise, group theory helps predict the changes in physical properties that occur when amaterial undergoes a phase transition, for example, from a cubic to a tetrahedralcrystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to achange from the high-symmetry paraelectric state to the lower symmetry ferroelectic

state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goesto zero frequency at the transition.

Such spontaneous symmetry breaking has found further application in elementary particl physics, where its occurrence is related to the appearance of Goldstone bosons.

Buckminsterfullerenedisplays

icosahedralsymmetry.

Ammonia, NH3. Itssymmetrygroup is of

order 6,generated by a120 rotation

and areflection.

Cubane C8H8features

octahedralsymmetry.

Hexaaquacopper(II)complex ion,[Cu(OH2)6]2+.Compared to a

perfectlysymmetrical shape,

the molecule isvertically dilated byabout 22% (Jahn-

Teller effect).

The (2,3,7)trianglegroup, a

hyperbolicgroup, acts

on this tilingof the

hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is inturn applied in error correction of transmitted data, and in CD players. Anotherapplication is differential Galois theory, which characterizes functions havingantiderivatives of a prescribed form, giving group-theoretic criteria for when solutions ofcertain differential equations are well-behaved. Geometric properties that remain stableunder group actions are investigated in (geometric) invariant theory.


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General linear group and representation theory

Two vectors (the left illustration) multiplied by matrices (the middle and rightillustrations). The middle illustration represents a clockwise rotation by 90, while the

right-most one stretches the x

-coordinate by factor 2.Matrix groups consist of matrices together with matrix multiplication. Thegeneral lineargroup GL(n, R ) consists of all invertiblen-by-n matrices with real entries. Its subgroupsare referred to asmatrix groups or linear groups . The dihedral group example mentionedabove can be viewed as a (very small) matrix group. Another important matrix group isthe special orthogonal groupSO(n). It describes all possible rotations inn dimensions.Via Euler angles, rotation matrices are used in computer graphics.

Representation theory is both an application of the group concept and important for adeeper understanding of groups. It studies the group by its group actions on other spaces.A broad class of group representations are linear representations, i.e. the group is actingon a vector space, such as the three-dimensional Euclidean spaceR 3. A representation ofG on ann-dimensional real vector space is simply a group homomorphism

: G GL(n, R)

from the group to the general linear group. This way, the group operation, which may beabstractly given, translates to the multiplication of matrices making it accessible toexplicit computations.

Given a group action, this gives further means to study the object being acted on On theother hand, it also yields information about the group. Group representations are anorganizing principle in the theory of finite groups, Lie groups, algebraic groups andtopological groups, especially (locally) compact groups.


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Galois groups

Galois groups have been developed to help solve polynomial equations by capturing theirsymmetry features. For example, the solutions of the quadratic equationax2 + bx + c = 0are given by

Exchanging "+" and "" in the expression, i.e. permuting the two solutions of theequation can be viewed as a (very simple) group operation. Similar formulae are knownfor cubic and quartic equations, but donot exist in general for degree 5 and higher.Abstract properties of Galois groups associated with polynomials (in particular theirsolvability) give a criterion for polynomials that have all their solutions expressible byradicals, i.e. solutions expressible using solely addition, multiplication, and roots similarto the formula above.

The problem can be dealt with by shifting to field theory and considering the splittingfield of a polynomial. Modern Galois theory generalizes the above type of Galois groupsto field extensions and establishesvia the fundamental theorem of Galois theorya precise relationship between fields and groups, underlining once again the ubiquity ofgroups in mathematics.

Finite groups

A group is called finite if it has a finite number of elements. The number of elements iscalled the order of the groupG. An important class is thesymmetric groups S N , thegroups of permutations of N letters. For example, the symmetric group on 3 lettersS 3 isthe group consisting of all possible swaps of the three letters ABC , i.e. contains theelements ABC , ACB , ..., up toCBA, in total 6 (or 3 factorial) elements. This class isfundamental insofar as any finite group can be expressed as a subgroup of a symmetricgroupS N for a suitable integer N (Cayley's theorem). Parallel to the group of symmetriesof the square above,S 3 can also be interpreted as the group of symmetries of anequilateral triangle.

The order of an element a in a group G is the least positive integer n such that a n = e,where a n represents

i.e. application of the operation to n copies of a. (If represents multiplication, then a n corresponds to the n th power of a.) In infinite groups, such an n may not exist, in whichcase the order of a is said to be infinity. The order of an element equals the order of thecyclic subgroup generated by this element.


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More sophisticated counting techniques, for example counting cosets, yield more precisestatements about finite groups: Lagrange's Theorem states that for a finite groupG theorder of any finite subgroup H divides the order ofG. The Sylow theorems give a partialconverse.

The dihedral group (discussed above) is a finite group of order 8. The order of r 1 is 4, asis the order of the subgroup R it generates. The order of the reflection elements f v etc. is2. Both orders divide 8, as predicted by Lagrange's Theorem. The groupsF p above haveorder p 1.

Classification of finite simple groups

Mathematicians often strive for a complete classification (or list) of a mathematicalnotion. In the context of finite groups, this aim quickly leads to difficult and profoundmathematics. According to Lagrange's theorem, finite groups of order p, a prime number,are necessarily cyclic (abelian) groupsZ p. Groups of order p2 can also be shown to be

abelian, a statement which does not generalize to order p3

, as the non-abelian group D4 oforder 8 = 23 above shows. Computer algebra systems can be used to list small groups, butthere is no classification of all finite groups. An intermediate step is the classification offinite simple groups. A nontrivial group is calledsimple if its only normal subgroups arethe trivial group and the group itself.The JordanHlder theorem exhibits finite simplegroups as the building blocks for all finite groups. Listing all finite simple groups was amajor achievement in contemporary group theory. 1998 Fields Medal winner RichardBorcherds succeeded to prove the monstrous moonshine conjectures, a surprising anddeep relation of the largest finite simple sporadic groupthe "monster group"withcertain modular functions, a piece of classical complex analysis, and string theory, atheory supposed to unify the description of many physical phenomena.

Groups with additional structure

Many groups are simultaneously groups and examples of other mathematical structures.In the language of category theory, they are group objects in a category, meaning thatthey are objects (that is, examples of another mathematical structure) which come withtransformations (called morphisms) that mimic the group axioms. For example, everygroup (as defined above) is also a set, so a group is a group object in the category of sets.


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Topological groups

The unit circle in the complex plane under complex multiplication is a Lie group and,therefore, a topological group. It is topological since complex multiplication and divisionare continuous. It is a manifold and thus a Lie group, because every small piece, such asthe red arc in the figure, looks like a part of the real line (shown at the bottom).

Some topological spaces may be endowed with a group law. In order for the group lawand the topology to interweave well, the group operations must be continuous functions,that is,g h, andg1 must not vary wildly ifg andh vary only little. Such groups arecalledtopological groups, and they are the group objects in the category of topologicalspaces. The most basic examples are the realsR under addition, (R \ {0}, ), and similarlywith any other topological field such as the complex numbers or p-adic numbers. All ofthese groups are locally compact, so they have Haar measures and can be studied viaharmonic analysis. The former offer an abstract formalism of invariant integrals.Invariance means, in the case of real numbers for example:

for any constantc. Matrix groups over these fields fall under this regime, as do adelerings and adelic algebraic groups, which are basic to number theory. Galois groups ofinfinite field extensions such as the absolute Galois group can also be equipped with atopology, the so-called Krull topology, which in turn is central to generalize the above


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sketched connection of fields and groups to infinite field extensions. An advancedgeneralization of this idea, adapted to the needs of algebraic geometry, is the talefundamental group.

Lie groups

Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e.they are spaces looking locally like some Euclidean space of the appropriate dimension.Again, the additional structure, here the manifold structure, has to be compatible, i.e. themaps corresponding to multiplication and the inverse have to be smooth.

A standard example is the general linear group introduced above: it is an open subset ofthe space of alln-by-n matrices, because it is given by the inequality

det ( A) 0,

where A denotes ann-by-n matrix.Lie groups are of fundamental importance in physics: Noether's theorem links continuoussymmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to constructsimple modelsimposing, say, axial symmetry on a situation will typically lead tosignificant simplification in the equations one needs to solve to provide a physicaldescription. Another example are the Lorentz transformations, which relatemeasurements of time and velocity of two observers in motion relative to each other.They can be deduced in a purely group-theoretical way, by expressing thetransformations as a rotational symmetry of Minkowski space. The latter servesin the

absence of significant gravitationas a model of space time in special relativity. The fullsymmetry group of Minkowski space, i.e. including translations, is known as the Poincargroup. By the above, it plays a pivotal role in special relativity and, by implication, forquantum field theories. Symmetries that vary with location are central to the moderndescription of physical interactions with the help of gauge theory.

GeneralizationsGroup-like structures

Totality Associativity Identity InversesGroup Yes Yes Yes Yes

Monoid Yes Yes Yes NoSemigroup Yes Yes No No

Loop Yes No Yes YesQuasigroup Yes No No No

Magma Yes No No NoGroupoid No Yes Yes Yes


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Category No Yes Yes No

In abstract algebra, more general structures are defined by relaxing some of the axiomsdefining a group. For example, if the requirement that every element has an inverse iseliminated, the resulting algebraic structure is called a monoid. The natural numbersN (including 0) under addition form a monoid, as do the nonzero integers undermultiplication (Z \ {0}, ). There is a general method to formally add inverses to elementsto any (abelian) monoid, much the same way as (Q \ {0}, ) is derived from (Z \ {0}, ),known as the Grothendieck group. Groupoids are similar to groups except that thecompositiona b need not be defined for alla andb. They arise in the study of morecomplicated forms of symmetry, often in topological and analytical structures, such as thefundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitraryn-ary one (i.e. an operation takingn arguments). With the proper generalization of the group axioms this gives rise to ann-arygroup. The table gives a list of several structures generalizing groups.


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Chapter 3

Group Action

Given an equilateral triangle, the counterclockwise rotation by 120 around the center ofthe triangle "acts" on the set of vertices of the triangle by mapping every vertex to

another one.In algebra and geometry, agroup action is a way of describing symmetries of objectsusing groups. The essential elements of the object are described by a set and thesymmetries of the object are described by the symmetry group of this set, which consistsof bijective transformations of the set. In this case, the group is also called apermutationgroup (especially if the set is finite or not a vector space) ortransformation group


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(especially if the set is a vector space and the group acts like linear transformations of theset).

A group action is a flexible generalization of the notion of a symmetry group in whichevery element of the group "acts" like a bijective transformation (or "symmetry") of some

set, without being identified with that transformation. This allows for a morecomprehensive description of the symmetries of an object, such as a polyhedron, byallowing the same group to act on several different sets, such as the set of vertices, the setof edges and the set of faces of the polyhedron.

If G is a group and X is a set then a group action may be defined as a grouphomomorphism fromG to the symmetric group of X . The action assigns a permutation of

X to each element of the group in such a way that

the permutation of X assigned to the identity element ofG is the identitytransformation of X ;

the permutation of X assigned to a productgh of two elements of the group is thecomposite of the permutations assigned tog andh.

Since each element ofG is represented as a permutation, a group action is also known asa permutation representation .

The abstraction provided by group actions is a powerful one, because it allowsgeometrical ideas to be applied to more abstract objects. Many objects in mathematicshave natural group actions defined on them. In particular, groups can act on other groupsor even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep

results in several fields.Definition

If G is a group and X is a set, then a (left ) group action ofG on X is a binary function

denoted

which satisfies the following two axioms:

1. (gh) x = g(h x) for allg, h inG and x in X ;2. e x = x for every x in X (wheree denotes the identity element of G).

The set X is called a (left ) G-set . The groupG is said to act on X (on the left).


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From these two axioms, it follows that for everyg inG, the function which maps x in Xto g x is a bijective map from X to X (its inverse being the function which maps x tog-1 x).Therefore, one may alternatively define a group action ofG on X as a grouphomomorphism fromG into the symmetric group Sym( X ) of all bijections from X to X .

In complete analogy, one can define aright group action ofG on X as a function X G X by the two axioms:

1. x(gh) = ( xg)h;2. xe = x.

The difference between left and right actions is in the order in which a product likegh acts on x. For a left actionh acts first and is followed byg, while for a right actiong actsfirst and is followed byh. From a right action a left action can be constructed bycomposing with the inverse operation on the group. Ifr is a right action, then

is a left action, since

and

Similarly, any left action can be converted into a right action. Therefore in the sequel weconsider only left group actions, since right actions add nothing new.

Examples

Thetrivial action for any groupG is defined byg x= x for allg inG and all x in X ;that is, the whole groupG induces the identity permutation on X .

Every groupG acts onG in two natural but essentially different ways:g x = gx forall x inG, org x = gxg1 for all x inG. The latter action is often called theconjugation action, and an exponential notation is commonly used for the right-

action variant: xg

= g1 xg; it satisfies ( x

g

)h

= xgh

. The symmetric group Sn and its subgroups act on the set { 1, ... ,n } by permutingits elements

The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.

The symmetry group of any geometrical object acts on the set of points of thatobject


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The automorphism group of a vector space (or graph, or group, or ring...) acts onthe vector space (or set of vertices of the graph, or group, or ring...).

The general linear group GL(n,R ), special linear group SL(n,R ), orthogonal groupO(n,R), and special orthogonal group SO(n,R ) are Lie groups which act onR n.

The Galois group of a field extension E /F acts on the bigger field E . So does

every subgroup of the Galois group. The additive group of the real numbers (R , +) acts on the phase space of "well- behaved" systems in classical mechanics (and in more general dynamicalsystems): ift is inR and x is in the phase space, then x describes a state of thesystem, andt x is defined to be the state of the systemt seconds later ift is positive ort seconds ago ift is negative.

The additive group of the real numbers (R , +) acts on the set of real functions of areal variable with (g f )( x) equal to e.g. f ( x + g), f ( x) +g, f ( xeg), f ( x)eg, f ( x + g)eg, or

f ( xeg) +g, but not f ( xeg + g) The quaternions with modulus 1, as a multiplicative group, act onR 3: for any

such quaternion , the mapping f (x) = z x z*

is acounterclockwise rotation through an angle about an axisv; z is the samerotation.

The isometries of the plane act on the set of 2D images and patterns, such as awallpaper pattern. The definition can be made more precise by specifying what ismeant by image or pattern, e.g. a function of position with values in a set ofcolors.

More generally, a group of bijectionsg: V V acts on the set of functions x: V W by (gx)(v) = x(g1(v)) (or a restricted set of such functions that is closedunder the group action). Thus a group of bijections of space induces a groupaction on "objects" in it.

Types of actions

The action ofG on X is called

transitive if X is non-empty and if equivalently1. for any x, y in X there exists ag inG such thatgx = y,2. Gx = X for all x in X ,3. Gx = X for some x in X .

Here,Gx = {g. x | g inG} is the orbit of x underG.

o sharply transitive if thatg is unique; it is equivalent to regularity defined below.

n -transitive if X has at leastn elements and for any pairwise distinct x1, ..., xn and pairwise distinct y1, ..., yn there is ag inG such thatg. xk = yk for 1 k n. A 2-transitive action is also calleddoubly transitive , a 3-transitive action is alsocalledtriply transitive , and so on. Such actions define 2-transitive groups, 3-transitive groups, and multiply transitive groups.


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o sharply n -transitive if there is exactly one suchg. faithful (oreffective ) if for any two distinctg, h inG there exists an x in X such

thatg x h x; or equivalently, if for anyg e inG there exists an x in X such thatg x x. Intuitively, different elements of G induce different permutations of X.

free (orsemiregular ) if for all x in X , g. x = h. x only ifg = h. Equivalently: if

there exists an x in X such thatg. x = x (that is, ifg has at least one fixed point),theng is the identity. regular (orsimply transitive ) if it is both transitive and free; this is equivalent to

saying that for any two x, y in X there exists precisely oneg inG such thatg x = y.In this case, X is known as a principal homogeneous space forG or as a G-torsor.

locally free if G is a topological group, and there is a neighbourhoodU ofe inG such that the restriction of the action toU is free; that is, ifg x = x for some x andsomeg inU theng = e .

irreducible if X is a nonzero module over a ring R, the action ofG is R-linear,and there is no nonzero proper invariant submodule.

Every free action on a non-empty set is faithful. A groupG acts faithfully on X if andonly if the homomorphismG Sym( X ) has a trivial kernel. Thus, for a faithful action,G is isomorphic to a permutation group on X ; specifically,G is isomorphic to its image inSym( X ).

The action of any groupG on itself by left multiplication is regular, and thus faithful aswell. Every group can, therefore, be embedded in the symmetric group on its ownelements, Sym(G) a result known as Cayley's theorem.

If G does not act faithfully on X , one can easily modify the group to obtain a faithfulaction. If we define N = {g inG : g x = x for all x in X }, then N is a normal subgroup of

G; indeed, it is the kernel of the homomorphismG Sym( X ). The factor groupG/ N actsfaithfully on X by setting (gN ) x = g x. The original action ofG on X is faithful if andonly if N = {e}.


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Orbits and stabilizers

In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedralgroup I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational)tetrahedral groupT of order 12, and the orbit space I /T (of order 60/12 = 5) is naturallyidentified with the 5 tetrahedra the cosetgT corresponds to which tetrahedrong sendsthe chosen tetrahedron to.

Consider a groupG acting on a set X . Theorbit of a point x in X is the set of elements of X to which x can be moved by the elements ofG. The orbit of x is denoted byGx:

The defining properties of a group guarantee that the set of orbits of (points x in) X underthe action ofG form a partition of X . The associated equivalence relation is defined by


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saying x ~ y if and only if there exists ag inG withg x = y. The orbits are then theequivalence classes under this relation; two elements x and y are equivalent if and only iftheir orbits are the same, i.e.Gx = Gy.

The set of all orbits of X under the action ofG is written as X /G (or, less frequently:G

\ X ), and is called thequotient of the action. In geometric situations it may be called the orbit space , while in algebraic situations it may be called the space of coinvariants , andwritten X G, by contrast with the invariants (fixed points), denoted X G: the coinvariants area quotient while the invariants are asubset. The coinvariant terminology and notation areused particularly in group cohomology and group homology, which use the samesuperscript/subscript convention.

If Y is a subset of X , we writeGY for the set {g y : y Y andg G}. We call the subsetY invariant under G if GY = Y (which is equivalent toGY Y ). In that case,G alsooperates onY . The subsetY is called fixed under G if g y = y for allg inG and all y inY .Every subset that's fixed underG is also invariant underG, but not vice versa.

Every orbit is an invariant subset of X on whichG acts transitively. The action ofG on X is transitive if and only if all elements are equivalent, meaning that there is only oneorbit.

For every x in X , we define thestabilizer subgroup of x (also called theisotropy group or little group ) as the set of all elements inG that fix x:

This is a subgroup ofG, though typically not a normal one. The action ofG on X is free ifand only if all stabilizers are trivial. The kernel N of the homomorphismG Sym( X ) isgiven by the intersection of the stabilizersG x for all x in X .

Orbits and stabilizers are closely related. For a fixed x in X , consider the map fromG to X given byg g x. The image of this map is the orbit of x and the coimage is the set of allleft cosets ofG x. The standard quotient theorem of set theory then gives a natural bijection betweenG /G x andGx. Specifically, the bijection is given byhG x h x. Thisresult is known as theorbit-stabilizer theorem .

If G and X are finite then the orbit-stabilizer theorem, together with Lagrange's theorem,gives

This result is especially useful since it can be employed for counting arguments.

Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups,G x andG y, are conjugate (in particular, they are isomorphic). More precisely: if y = g x,


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thenG y = gG x g1. Points with conjugate stabilizer subgroups are said to have the sameorbit-type .

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

where X g is the set of points fixed byg. This result is mainly of use whenG and X arefinite, when it can be interpreted as follows: the number of orbits is equal to the averagenumber of points fixed per group element.

The set of formal differences of finiteG-sets forms a ring called the Burnside ring, whereaddition corresponds to disjoint union, and multiplication to Cartesian product.

A G-invariant element of X is x X such thatg x = x for allg G. The set of all such x is denoted X G and called theG-invariants of X . When X is aG-module, X G is the zerothgroup cohomology group ofG with coefficients in X , and the higher cohomology groupsare the derived functors of the functor ofG-invariants.

Group actions and groupoids

The notion of group action can be put in a broader context by using the associated `actiongroupoid' associated to the group action, thus allowing techniques fromgroupoid theory such as presentations and fibrations. Further the stabilisers of the actionare the vertex groups, and the orbits of the action are the components, of the action

groupoid.

This action groupoid comes with a morphism which is a `coveringmorphism of groupoids'. This allows a relation between such morphisms and coveringmaps in topology.

Morphisms and isomorphisms between G-sets

If X andY are twoG-sets, we define amorphism from X toY to be a function f : X Y such that f (g x) =g f ( x) for allg inG and all x in X . Morphisms ofG-sets are also calledequivariant maps orG-maps .

If such a function f is bijective, then its inverse is also a morphism, and we call f anisomorphism and the twoG-sets X andY are calledisomorphic ; for all practical purposes,they are indistinguishable in this case.

Some example isomorphisms:

8/12/2019 Appli

applications of symmetry in mathematics, physics & chemistry

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