ग्रुप सिद्धान्त

**ग्रुप सिद्धान्त** गणितयागु छगु ख्यः ख। थुकिगु छ्येलेज्या भौतिकशास्त्र व रसायनशास्त्रय् जुइ। 

पुचः यागु छ्येलेज्या गणितय् यक्व थासे अप्वयाना इन्टर्नल सिमेट्रि क्याप्चर यायेत अटोमर्फिज ग्रुप यागु रुपे जुइ। An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group.

In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.

Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.

In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.

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

An understanding of group theory is also important in physics and chemistry and material science. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.
:Physics examples: Standard Model, Gauge theory

## इतिहास
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.

An early source occurs in the problem of forming an mth-degree equation having as its roots *m* of the roots of a given
nth-degree equation (m ). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.

A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (*résolvantes, réduites*) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a *Calcul des Combinaisons*. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.

Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called
intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation
under the name *l'assieme delle permutazioni*. He also published a letter from Abbati to himself, in which the group idea is prominent.

Galois found that if r_1, r_2, \ldots, r_n are the n roots of an equation, there is always a group of permutations of the r's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).

Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose *Traité des Substitutions* is a classic; and to Eugen Netto (1882), whose *Theory of Substitutions and its Applications to Algebra* was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.

It was Walther von Dyck who, in 1882, gave the modern definition of a group.

The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy. 

The classification of finite simple groups is a vast body of work from the mid 20th century, which is thought to classify all the finite simple groups. 

Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others.

## ग्रुप सिद्धान्त विचातः
### ग्रुपयागु अर्थ
:*मू पौ: [[पुचः (गणित)]]*
A group *(G, *)* is a set *G* with a binary operation * : *G* × *G* → *G* (one that assigns each ordered pair (a,b) in G an element in G denoted by a*b) that satisfies the following 3 axioms:

* *[[Associativity]]*: For all *a*, *b* and *c* in *G*, (*a* * *b*) * *c* = *a* * (*b* * *c*). 	 
* *[[Identity element]]*: There is an element *e* in *G* such that for all *a* in *G*, *e* * *a* = *a* * *e* = *a*. 	 
* *[[Inverse element]]*: For each *a* in *G*, there is an element *b* in *G* such that *a* * *b* = *b* * *a* = *e*, where *e* is an identity element.

### Order of groups and elements
The **[[Order (group theory)|order of a group]] ''G*** is the number of elements in the set *G*. If the order is not finite, then the group is an *infinite group*. 

The *'order of an element *a*** in a group ''G'' is the least positive integer ''n'' such that ''a<sup>n</sup>=e'', where ''a<sup>n</sup>'' is multiplication of ''a'' by itself ''n'' times.

For finite groups, it can be shown that the order of every element in the group must [[divisor|divide]] the order of the group.

=== Subgroups ===
A set ''H'' is a **subgroup** of a group ''G'' if it is a subset of ''G'' and is a group using the operation defined on ''G''. In other words, ''H'' is a subgroup of (''G'', *) if the restriction of * to ''H'' is a group operation on ''H''.

If ''G'' is a finite group, the order of ''H'' [[divisor|divides]] the order of ''G''.

A subgroup ''H'' is a [[normal subgroup]] of ''G'' if for all ''h'' in ''H'' and ''g'' in ''G'', ''ghg''<sup>-1</sup> is also in ''H''. An alternate definition is that a subgroup is normal if its left and right [[coset]]s coincide. 

Normal subgroups are useful because they can be used to create [[quotient group]]s.

=== Special classes of groups ===
A group is **abelian** (or **commutative**) if the operation is [[commutative]] (that is, for all ''a'', ''b'' in ''G'', ''a'' * ''b'' = ''b'' * ''a''). A **non-abelian** group is a group that is not abelian. The term "abelian" is named after the mathematician [[Niels Abel]].

A **cyclic group** is a group that is [[generating set of a group|generated]] by a single element.

A **simple group** is a group that has no nontrivial normal subgroups.

A **solvable group** , or soluble group, is a group that has a normal series whose quotient groups are all abelian. The fact that ''S''<sub>5</sub>, the [[symmetric group]] in 5 elements, is not solvable proves that some [[quintic equation|quintic]] polynomials cannot be solved by radicals.

=== Operations on groups ===
A **homomorphism** is a map between two groups that preserves the structure imposed by the operator. If the map is bijective, then it is an **isomorphism**. An isomorphism from a group to itself is an **automorphism**. The set of all automorphisms of a group is a group called the automorphism group. The [[kernel of a homomorphism]] is a normal subgroup of the group.

A **group action** is a map involving a group and a set, where each element in the group defines a bijective map on a set. Group actions are used to prove the [[Sylow theorems]] and to prove that the [[center (group theory)|center]] of a [[p-group]] is nontrivial.

== Some useful theorems ==
* Some basic results in [[elementary group theory]]
* **Lagrange's theorem**: if ''G'' is a finite group and ''H'' is a subgroup of ''G'', then the order (that is, the number of elements) of ''H'' divides the order of ''G''.
* **Cayley's Theorem**: every group ''G'' is isomorphic to a subgroup of the symmetric group on ''G''.
* **Sylow theorems**: perhaps the most useful of the group theorems. Among them, that if ''p''<sup>''n''</sup> (and ''p'' prime) divides the order of a finite group ''G'', then there exists a subgroup of order ''p''<sup>''n''</sup>.
* The **Butterfly lemma** is a technical result on the [[lattice of subgroups]] of a group.
* The **Fundamental theorem on homomorphisms** relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
* **Jordan-Hölder theorem**: any two composition series of a given group are equivalent.
* **Krull-Schmidt theorem**: a group G, subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite product of indecomposable subgroups.
* **Burnside's lemma''': the number of orbits of a group action on a set equals the average number of points fixed by each element of the group

## Miscellany

James Newman summarized group theory as follows:

:The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.

One application of group theory is in musical set theory.

Group theory is also very important to the field of chemistry, where it is used to assign symmetries to molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.

In Philosophy, Ernst Cassirer related the theory of group to the theory of perception as described by Gestalt Psychology; Perceptual Constancy is taken to be analogous to the invariants of group theory.

## लिधंसा

*   A standard modern reference.
*  An inexpensive and fairly readable textbook (somewhat outdated in emphasis, style, and notation).
*  Pleasant to read book that explains the importance of group theory and how its symmetries lead to parallel symmetries in the world of physics and other sciences. Really helps congeal the importance of group theory as a practical tool).

## पिनेयागु स्वापूतः
* एब्स्ट्र्याक्ट ग्रुप कन्सेप्टयागु इतिहास
* Higher dimensional group theory

पुचः:गणित