Maths Literature Survey

8/9/2019 Maths Literature Survey

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GROUPS EXAMPLES AND APPLICATIONS

A periodic wallpaper pattern gives rise to a wallpaper group .

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

Examples and applications of groups abound. A starting point is the groupZ of integers withaddition as group operation, introduced above. If instead of addition multiplication is considereone obtains multiplicative groups. These groups are predecessors of important constructions abstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects are ofteexamined by associating groups to them and studying the properties of the corresponding groupFor example, Henri Poincar founded what is now called algebraic topology by introducing thfundamental group. By means of this connection, topological properties such as proximity ancontinuity translate into properties of groups.For example, elements of the fundamental group arepresented by loops. The second image at the right shows some loops in a plane minus a poinThe blue loop is considered null-homotopic (and thus irrelevant), because it can be continuousshrunk to a point.


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The presence of the hole prevents the orange loop from being shrunk to a point. The fundamengroup of the plane with a point deleted turns out to be infinite cyclic, generated by the orangloop (or any other loop winding once around the hole). This way, the fundamental group detecthe hole.

In more recent applications, the influence has also been reversed to motivate geometriconstructions by a group-theoretical background. In a similar vein, geometric group theoemploys geometric concepts, for example in the study of hyperbolic groups.Further branchcrucially applying groups include algebraic geometry and number theory.

In addition to the above theoretical applications, many practical applications of groups exisCryptography relies on the combination of the abstract group theory approach together witalgorithmical knowledge obtained in computational group theory, in particular whenimplemented for finite groups.Applications of group theory are not restricted to mathematicsciences 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 given groustructure. In some cases, such as with the rationals, both addition and multiplication operationgive rise to group structures. Such number systems are predecessors to more general algebrastructures known as rings and fields. Further abstract algebraic concepts such as modules, vectspaces and algebras also form groups.

Integers

The group of integersZ under addition, denoted (Z , +), has been described above. The integers,with the operation of multiplication instead of addition, (Z , ) donot form a group. The closure,associativity and identity axioms are satisfied, but inverses do not exist: for example,a = 2 is aninteger, but the only solution to the equationa b = 1 in this case isb = 1/2, which is a rationalnumber, but not an integer. Hence not every element of Z has a (multiplicative) inverse.


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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 such fractionsis commonly denotedQ . There is still a minor obstacle for (Q , ), the rationals withmultiplication, being a group: because the rational number 0 does not have a multiplicativinverse (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 an abeliangroup under multiplication, denoted (Q \ {0}, ).Associativity and identity element axiomsfollow from the properties of integers. The closure requirement still holds true after removinzero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is

b/a , therefore the axiom of the inverse element is satisfied.

The rational numbers (including 0) also form a group under addition. Intertwining addition anmultiplication operations yields more complicated structures called rings andif division is

possible, such as inQ fields, which occupy a central position in abstract algebra. Grouptheoretic arguments therefore underlie parts of the theory of those entities.

Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group of integersmodulo p.[38] Its elements are integers not divisible by p, considered modulo p, i.e. two numbersare considered equivalent if their difference is divisible by p. For example, if p = 5, there areexactly four group elements 1, 2, 3, 4: multiples of 5 are excluded and 6 and 4 are bothequivalent to 1 etc. The group operation is given by multiplication. Therefore, 4 4 = 1, becauthe usual product 16 is equivalent to 1, for 5 divides 16 1 = 15, denoted

16 1 (mod 5).


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The primality of p ensures that the product of two integers neither of which is divisible by p isnot divisible by p either, hence the indicated set of classes is closed under multiplication.Theidentity element is 1, as usual for a multiplicative group, and the associativity follows from thcorresponding property of integers. Finally, the inverse element axiom requires that given ainteger a not divisible by p, there exists an integer b 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 greatest commondivisor gcd(a , p) equals 1.In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2,as 3 2 = 6 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar (Q \{0}, ) above, because it turns out to be the multiplicative group of nonzero elements in thfinite fieldF p, denotedF p

. These groups are crucial to public-key cryptography.

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 grou p is a group all of whose elements are powers (when the group operation is written

additively, the term 'multiple' can be used) of a particular elementa .In multiplicative notation,the elements of the group are:

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


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wherea2 meansa a , and a 3 stands for a 1 a 1 a 1 =(a a a)1 etc.Such an elementa iscalled a generator or a primitive element of the group.

A typical example for this class of groups is the group of n-th complex roots of unity, given by

complex numbers z satisfying zn = 1 (and whose operation is multiplication).Any cyclic groupwith n elements is isomorphic to this group. Using some field theory, the groupF p

can beshown 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-zerelementa , all the powers of a are distinct; despite the name "cyclic group", the powers of theelements do not cycle. An infinite cyclic group is isomorphic to (Z , +), the group of integersunder addition introduced above.As these two prototypes are both abelian, so is any cyclic grou

The study of abelian groups is quite mature, including the fundamental theorem of finitelgenerated abelian groups; and reflecting this state of affairs, many group-related notions, such center and commutator, describe the extent to which a given group is not abelian.

Symmetry groups

S ymmetry grou ps are groups consisting of symmetries of given mathematical objectsbe they of geometric nature, such as the introductory symmetry group of the square, or of algebraic natursuch as polynomial equations and their solutions.Conceptually, group theory can be thought of the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical oanalytical objects. A group is said to act on another mathematical object X if every groupelement 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 thhighlighted warped triangles (and the other ones, too). By a group action, the group pattern connected to the structure of the object 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 describe molecul

symmetries and crystal symmetries. These symmetries underlie the chemical and physic behavior of these systems, and group theory enables simplification of quantum mechanicanalysis of these properties.For example, group theory is used to show that optical transition between certain quantum levels cannot occur simply because of the symmetry of the statinvolved.

Not only are groups useful to assess the implications of symmetries in molecules, bsurprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Tell

effect is a distortion of a molecule of high symmetry when it adopts a particular ground state lower symmetry from a set of possible ground states that are related to each other by thsymmetry operations of the molecule.

Likewise, group theory helps predict the changes in physical properties that occur when material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline formAn example is ferroelectric materials, where the change from a paraelectric to a ferroelectrstate occurs at the Curie temperature and is related to a change from the high-symmetr paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called so phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

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


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Buckminsterfullerene displaysicosahedralsymmetry.

Ammonia, NH3.Its symmetrygroup is of order 6, generated by a120 rotation anda reflection.

Cubane C8H8 featuresoctahedralsymmetry.

Hexaaquacopper(II) complex ion,[Cu(OH2)6]2+.Compared to a perfectlysymmetricalshape, themolecule isvertically dilated by about 22%(Jahn-Teller

effect).

The (2,3,7)triangle group, ahyperbolic group,acts on this tilingof the hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in tuapplied in error correction of transmitted data, and in CD players.Another application idifferential Galois theory, which characterizes functions having antiderivatives of a prescribeform, giving group-theoretic criteria for when solutions of certain differential equations are we behaved.Geometric properties that remain stable under group actions are investigated

(geometric) invariant theory.


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General linear group and representation theory

Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). Thmiddle illustration represents a clockwise rotation by 90, while the right-most one stretches th

x-coordinate by factor 2.

Matrix groups consist of matrices together with matrix multiplication. The general linear grou p

GL(n, R ) consists of all invertiblen-by-n matrices with real entries. Its subgroups are referred toas matri x grou ps or linear grou ps. The dihedral group example mentioned above can be viewedas a (very small) matrix group. Another important matrix group is the special orthogonal grouSO(n). It describes all possible rotations inn dimensions. Via Euler angles, rotation matrices areused in computer graphics.

Re pre sentation theory is both an application of the group concept and important for a deeper understanding of groups.It studies the group by its group actions on other spaces. A broad claof group representations are linear representations, i.e. the group is acting on a vector space, suas the three-dimensional Euclidean spaceR 3. A representation of G on an n-dimensional realvector 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 babstractly given, translates to the multiplication of matrices making it accessible to expliccomputations.

Given a group action, this gives further means to study the object being acted on.On the othhand, it also yields information about the group.


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Group representations are an organizing principle in the theory of finite groups, Lie groupalgebraic groups and topological groups, especially (locally) compact groups.

Galois groups

Galoi s grou ps have been developed to help solve polynomial equations by capturing theirsymmetry features. For example, the solutions of the quadratic equationa x2 + b x + c = 0 aregiven by

Exchanging "+" and "" in the expression, i.e. permuting the two solutions of the equation ca be viewed as a (very simple) group operation. Similar formulae are known for cubic and quarequations, but donot exist in general for degree 5 and higher. Abstract properties of Galoisgroups associated with polynomials (in particular their solvability) give a criterion fo polynomials that have all their solutions expressible by radicals, i.e. solutions expressible usinsolely addition, multiplication, and roots similar to the formula above.

The problem can be dealt with by shifting to field theory and considering the splitting field of

polynomial. Modern Galois theory generalizes the above type of Galois groups to fieextensions and establishesvia the fundamental theorem of Galois theorya preciserelationship between fields and groups, underlining once again the ubiquity of groups imathematics.


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UNIT-V

GROUPS

In mathematics, agroup is a set endowed with a binary operation satisfying certain axioms,detailed below. For example, the set of integers withaddition as the binary operation is a group.Group theory is the branch of mathematics which studies groups.

Group theory originated with the work of variste Galois, in 1830, on the problem of when aalgebraic equation is soluble by radicals. Before this work, groups were mainly studied in termof permutations. Some aspects of abelian group theory were also known in the theory oquadratic forms.

Many of the structures investigated in mathematics turn out to be groups. These include familinumber systems, such as the integers, the rational numbers, the real numbers, and the complenumbers under addition, as well as the non-zero rationals, reals, and complex numbers, undmultiplication. Other important examples are the group of non-singular matrices undemultiplication and the group of invertible functions under composition. Group theory allows oto study such structures in a general setting.

Group theory has extensive applications in mathematics, science, and engineering. Manalgebraic structures such as fields and vector spaces are based on groups, and group theo provides an important tool for studying symmetry, since the symmetries of any object formgroup. Groups are thus pertinent to branches of sciences involving symmetry principles, such relativity, quantum mechanics, particle physics, chemistry, computer graphics, and others.

H ISTORY

The modern concept of an abstract group developed out of several fields of mathematics.Thoriginal motivation for group theory was the quest for solutions of polynomial equations degree higher than 4. The 19th-century French mathematician variste Galois, extending priwork of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of particular polynomial equation in terms of the symmetry group of its roots (solutions). Thelements of such a Galois group correspond to certain permutations of the roots.


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At first, Galois' ideas were rejected by his contemporaries, and published only posthumously.More general permutation groups were investigated in particular by AugustLouis Cauchy. Arthur Cayley'sOn the theory of grou ps, a s de pending on the s ymbolic e quation

n = 1 (1854) gives the first abstract definition of a finite group.

Geometry was a second field in which groups were used systematically, especially symmetrgroups as part of Felix Klein's 1872 Erlangen program.

After novel geometries such as hyperbolic and projective geometry had emerged, Klein usegroup theory to organize them in a more coherent way. Further advancing these ideas, SophuLie founded the study of Lie groups in 1884.

The third field contributing to group theory was number theory. Certain abelian group structurhad been used implicitly in Carl Friedrich Gauss' number-theoretical work Di squi sitione s

Arithmeticae (1798), and more explicitly by Leopold Kronecker.In 1847, Ernst Kummer ledearly attempts to prove Fermat's Last Theorem to a climax by developing groups describinfactorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started with CamilJordan'sTrait de s sub stitution s et de s quation s algbri que s (1870).Walther von Dyck (1882)gave the first statement of the modern definition of an abstract group.As of the 20th centurgroups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius anWilliam Burnside, who worked on representation theory of finite groups, Richard Brauermodular representation theory and Issai Schur's papers.The theory of Lie groups, and mogenerally locally compact groups was pushed by Hermann Weyl, lie Cartan and many otherIts algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevall(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 group theorists suas Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of acollaboration that, with input from numerous other mathematicians, classified all finite simpgroups in 1982. This project exceeded previous mathematical endeavours by its sheer size, both length of proof and number of researchers.


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Research is ongoing to simplify the proof of this classification.These days, group theory is stillhighly active mathematical branch crucially impacting many other fields.

B ASIC DE F INITIONS

A group (G, * ) is a setG along with a function * :G G G, satisfying the group axioms below. The function * is called abinary o perator . Here "a * b" represents the result of applyingthe function * to the ordered pair (a , b) of elements inG. The group axioms are the following:

y A ssociativity : For everya , b andc in G, (a * b) * c = a * (b * c).

y N eutral element : There is an elemente in G such that for everya in G, e * a = a * e = a.

y I nver se element : For everya in G, there is an elementb in G such thata * b = b * a = e,wheree is the neutral element from the previous axiom.

One often also sees the axiom:

y C lo sure : For alla andb in G, a * b belongs toG.

Since the definition of group given here uses the notion of binary operation, closure iautomatically satisfied and hence would be superfluous as an axiom. When determining if given * is a group operation, one nevertheless checks that * satisfies closure as part of verifyin

that it is, in fact, a binary operation.

The neutral element of a group is often called the identity element if the operation is written multiplicative notation, while it is called the zero element or null element if the operation written in additive notation.

If a group has both a left neutral element (say,e1) and a right neutral element (say,e2), then theymust be identical (becausee1 = e1 * e2 = e2). It follows that the neutral elemente of the secondgroup axiom is unique, that is, a group has only one neutral element. This is why the third grouaxiom refers tothe neutral element, even though the second axiom merely asserts that there is atleast one neutral element.

Theorder of a groupG, denoted by |G | or o(G), is the number of elements of the setG. A groupis called finite if it has finitely many elements, that is if the setG is a finite set.


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When there is no ambiguity, the group (G, * ) is often denoted simply as "G", leaving theoperation * unmentioned. But different operations on the same set would define different group

The operation in a group need not be commutative, that is there may exist elementsa ,b such that

a * b b * a . A groupG is said to beabelian (after the mathematician Niels Abel) (or commutative ) if for everya , b in G, a * b = b * a. Groups lacking this property are callednon-abelian .

ALTERNATIVE AXIOMATIZATIONS

The axioms given above in the definition of group are stronger than what is strictly requireSufficient are associativity, the existence of a right neutral element (that is, there is an elemente

such that x * e = x for all x), and the existence of right inverses with respect to this right neutralelement (that is, for each x, there is a y such that x * y = e). It follows from these that the postulated right neutral elemente is also a left neutral element, and hence, as above, is unique.Further, it follows that each right inverse is also a left inverse. Thus, the axiomatization giveabove is not strictly minimal in the logical sense; however, it is customary. One reason for thcustom is that the axioms as given are easily remembered and checked in practice. Anothereason is that subsets or variants of the axioms define other useful algebraic structures e.g.groupoids and semigroups.

Groups can be axiomatized in ways other than the one presented above. For instance, a group isset G closed under:

y An associative binary operation, here denoted by concatenation, and a unary operationdenoted by the superscript -1 such that x-1 x y = y x-1 x = y is an axiom. It follows that x-1 x isa constant, and hence, is the neutral element.

y An associative binary operation, here denoted by concatenation, such that for eacha ,b

G, there exist x, y G such thata x = b and ya = b. Equivalently, a group is an associativequasigroup. The existence of the neutral element follows easily.

y Two binary operations, here denoted by infix "/" and "\", with axioms y = x/( y\ x) =( x/ y)\ x, and ( x/ y)\ z = x/( y\ z). One may then define an associative binary operation and aunary operation as above in terms of \ and / so that x/ y = x y-1 and x\ y = x-1 y.


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NOTATION F OR GROUPS

Convention Multiplication Addition

Operation x * y or x y x + y

Identity e or 1 0

Powers xn n x

Inverse x1 x

Direct sum G H G H

Usually the operation, whatever itreally is, is thought of as an analogue of multiplication, andthe group operations are thereforewritten multi plicatively . That is:

y We write "a b" or even "ab " for a * b and call it the product of a andb;

y We write "1" (or "e") for the neutral element and call it theunit element ;

y We write "a 1 " for the inverse of a and call it thereci procal of a .

However, sometimes the group operation is thought of as analogous to addition andwrittenadditively :

y We write "a + b" for a * b and call it the sum of a andb;

y We write "0" for the neutral element and call it the zero element ;

y We write "a" for the inverse of a and call it theo ppo site of a .

Usually, only abelian (commutative) groups are written additively, although they may also bwritten multiplicatively. When being noncommittal, one can use the notation (with "*") anterminology that was introduced in the definition, using the notationa 1 for the inverse of a .

If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products { xs : s in S }; similarly the notationSx = { sx : s in S }; and for two subsetsS andT of G,


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this case isb = 1/2 . We cannot chooseb = 1/2 because 1/2 is not an integer. (Inverseelement fail s)

Since not every element of (Z ,) has an inverse, (Z ,) isnot a group. The most we can say is thatit is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbersQ , that is the set of numbersa/b such thata and b areintegers andb is nonzero, and the operation multiplication, denoted by "". Since the rationalnumber 0 does not have a multiplicative inverse, (Q ,), like (Z,), is not a group.

However, if we instead use the setQ \ {0} instead of Q , that is include every rational number

e xce pt zero, then (Q \ {0},)doe s form an abelian group (written multiplicatively). The inverse of a/b is b/a , and the other group axioms are simple to check. We don't lose closure by removingzero, because the product of two nonzero rationals is never zero.

Just as the integers form a ring, the rational numbers form the algebraic structure of a fieldallowing the operations of addition, subtraction, multiplication and division. In fact, the nonzeelements of any given field form a group under multiplication, called themulti plicative grou p of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, consider three colored blocks (red, green, and blue), initial placed in the order RGB. Leta be the action "swap the first block and the second block", and letb be the action "swap the second block and the third block".

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

In multiplicative form, we traditionally write x y for the combined action "first do y, then do x";so thatab is the action RGB RBG BRG, i.e., "take the last block and move it to the front".


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If we writee for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:

y e : RGB RGB

y a : RGB GRBy b : RGB RBG

y ab : RGB BRG

y ba : RGB GBR

y aba : RGB BGR

Note that the actionaa has the effect RGB GRB RGB, leaving the blocks as they were; sowe can writeaa = e. Similarly,

y bb = e,

y (aba )(aba ) = e, and

y (ab )(ba ) = (ba )(ab ) = e;

so each of the above actions has an inverse.

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

y (ab )a = a(ba ) = aba , andy (ba )b = b(ab ) = bab .

This group is called the s ymmetric grou p on 3 letter s, or S 3. It has order 6 (or 3 factorial), and isnon-abelian (since, for example,ab ba ). SinceS 3 is built up from the basic actionsa andb, wesay that the set {a ,b} generate s it.

Every group can be expressed in terms of permutation groups likeS 3; this result is Cayley'stheorem and is studied as part of the subject of group actions.

SIMPLE T H EOREMS

y A group has exactly one identity element.

y Every element has exactly one inverse.


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P roof : Suppose bothb and c are inverses of x. Then, by the definition of an inverse, xb = b x = e and xc = c x = e. But then:

(multiplying on the left byb)

(usingb x = e)

(neutral element axiom)

Therefore the inverse is unique.

The first two properties actually follow from associative binary operations defined on a seGiven a binary operation on a set, there is at most one identity and at most one inverse for anelement.

y You can perform division in groups; that is, given elementsa and b of the groupG, there isexactly one solution x in G to the equation x * a = b and exactly one solution y in G to theequationa * y = b.

y

The expression "a1 * a2 * *a n" is unambiguous, because the result will be the same no matter where we place parentheses.

y (S ock s and shoe s) The inverse of a product is the product of the inverses in the opposite order: (a * b)1 = b1 * a 1 .

P roof : We will demonstrate that (ab)(b-1a-1) = (b-1a-1)(ab) = e, as required by the definition of aninverse.

= (associativity)

= (definition of inverse)

= (definition of neutral element)


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= (definition of inverse)

And similarly for the other direction.

These and other basic facts that hold for all individual groups form the field of elementary groutheory.

CONSTRUCTING NEW GROUPS F ROM GIVEN ONES

1. If a subset H of a group (G,*) together with the operation * restricted on H is itself a group, thenit is called a subgrou p of (G,*).

2. Thedirect product of two groups (G,*) and ( H , ) is the Cartesian product setG H together with

the operation ( g 1,h1)( g 2,h2) = ( g 1* g 2,h1 h2). The direct product can also be defined with anynumber of terms, finite or infinite, by using the cartesian product and defining the operatiocoordinate-wise.

3. The semidirect product of two groups N and H with respect to a group homomorphism : H Aut( N ) is a new group ( N H , *), with * defined as

(n1, h1) * (n2, h2) = (n1 (h1) (n2), h1 h2)

4. The direct e xternal sum of a family of groups is the subgroup of the product constituted byelements that have a finite number of non-identity coordinates. If the family is finite the diresum and the product are of course the same.

5. Given a groupG and a normal subgroup N , thequotient grou p is the set of cosets of G / N together with the operation ( g N )(h N )= gh N .

GENERALIZATIONS

In abstract algebra, we get some related structures which are similar to groups by relaxing somof the axioms given at the top of the article.

y If we eliminate the requirement that every element have an inverse, then we get a monoid.y If we additionally do not require an identity either, then we get a semigroup.


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y Alternatively, if we relax the requirement that the operation be associative while still requirinthe possibility of division, then we get a loop.

y If we additionally do not require an identity, then we get a quasigroup.

y If we don't require any axioms of the binary operation at all, then we get a magma.

Groupoids, which are similar to groups except that the compositiona * b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological andanalytical structures. They are special sorts of categories.

Supergroups and Hopf algebras are other generalizations.

Lie groups, algebraic groups and topological groups are examples of group objects: group-likstructures sitting in a category other than the ordinary category of sets.

Abelian groups form the prototype for the concept of an abelian category, which has applicatioto vector spaces and beyond.

Formal group laws are certain formal power series which have properties much like a grouoperation.

APPLICATIONS O F GROUP T H EORY

Applications of group theory abound. Almost all structures inabstract algebraare special casesof groups.Rings, for example, can be viewed asabelian groups(corresponding to addition)together with a second operation (corresponding to multiplication). Therefore group theoretarguments underlie large parts of the theory of those entities.

Galois theoryuses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). Thefundamental theoremof Galois theoryprovides a link betweenalgebraic field extensionsand group theory. It gives an

effective criterion for the solvability of polynomial equations in terms of the solvability of thcorrespondingGalois group. For example,S 5, thesymmetric groupin 5 elements, is not solvablewhich implies that the generalquintic equationcannot be solved by radicals in the way equationsof lower degree can. The theory, being one of the historical roots of group theory, is stilfruitfully applied to yield new results in areas such asclass field theory.


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Algebraic topologyis another domain which prominentlyassociatesgroups to the objects thetheory is interested in. There, groups are used to describe certain invariants of topological spaces.They are called "invariants" because they are defined in such a way that they do not change if tspace is subjected to somedeformation. For example, thefundamental group"counts" how many paths in the space are essentially different. ThePoincar conjecture, proved in 2002/2003 byGrigori Perelmanis a prominent application of this idea. The influence is not unidirectional,though. For example, algebraic topology makes use of Eilenberg-MacLane spaceswhich arespaces with prescribedhomotopy groups. Similarlyalgebraic K-theorystakes in a crucial way onclassifying spacesof groups. Finally, the name of thetorsion subgroupof an infinite groupshows the legacy of topology in group theory.

A torus. Its abelian group structure is induced from the mapC C /Z+ Z, where is a parameter

Thecyclic group Z/26 underlies Caesar's cipher.

Algebraic geometryand cryptographylikewise uses group theory in many ways.Abelianvarietieshave been introduced above. The presence of the group operation yields additionainformation which makes these varieties particularly accessible. They also often serve as a tefor new conjectures.The one-dimensional case, namelyelliptic curvesis studied in particular detail. They are both theoretically and practically intriguing. Very large groups of prime ord


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constructed inElliptic-Curve Cryptographyserve for public key cryptography. Cryptographicalmethods of this kind benefit from the flexibility of the geometric objects, hence their groustructures, together with the complicated structure of these groups, which make thediscretelogarithmvery hard to calculate. One of the earliest encryption protocols,Caesar's cipher , mayalso be interpreted as a (very easy) group operation. In another direction,toric varietiesarealgebraic varietiesacted on by atorus. Toroidal embeddings have recently led to advances inalgebraic geometry, in particular resolution of singularities.

Algebraic number theoryis a special case of group theory, thereby following the rules of thelatter. For example,Euler's product formula

capturesthe factthat any integer decomposes in a unique way into primes. The failure of thisstatement for more general ringsgives rise toclass groupsand regular primes, which feature inKummer'streatment of Fermat's Last Theorem.

y The concept of theLie group(named after mathematicianSophus Lie) is important in thestudy of differential equationsand manifolds; they describe the symmetries of continuousgeometric and analytical structures. Analysis on these and other groups is calledharmonic analysis. Haar measures, that is integrals invariant under the translation in a Liegroup, are used for pattern recognitionand other image processingtechniques.

y In combinatorics, the notion of permutationgroup and the concept of group action areoften used to simplify the counting of a set of objects; see in particular Burnside's lemma.


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The circle of fifths may be endowed with a cyclic group structure

y The presence of the 12- periodicityin thecircle of fifthsyields applications of elementary

group theoryin musical set theory.

y An understanding of group theory is also important in physics and chemistry and materiscience. In physics, groups are important because they describe the symmetries which thlaws of physics seem to obey. Physicists are very interested in group representationsespecially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include:Standard Model,Gauge theory, Lorentz group, Poincar group

y In chemistry, groups are used to classify crystal structures, regular polyhedra, and thesymmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarityand chirality), spectroscopic properties (particularlyuseful for Raman spectroscopyand Infrared spectroscopy), and to construct molecular orbitals.

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Maths Literature Survey