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.
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.
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 th-degree equation having as its roots m of the roots of a given th-degree equation ( ). 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 are the roots of an equation, there is always a group of permutations of the '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.
ग्रुप सिद्धान्त विचातःEdit
- मू पौ: पुचः (गणित)
- 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 elementsEdit
The 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 an=e, where an 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 divide the order of the group.
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 divides the order of G.
Normal subgroups are useful because they can be used to create quotient groups.
Special classes of groupsEdit
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 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 S5, the symmetric group in 5 elements, is not solvable proves that some quintic polynomials cannot be solved by radicals.
Operations on groupsEdit
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 of a p-group is nontrivial.
Some useful theoremsEdit
- 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 pn (and p prime) divides the order of a finite group G, then there exists a subgroup of order pn.
- 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
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.
- Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8. A standard modern reference.
- Scott, W. R.  (1987). Group Theory. New York: Dover. ISBN 0-486-65377-3. An inexpensive and fairly readable textbook (somewhat outdated in emphasis, style, and notation).
- Livio, M. (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. ISBN 0-7432-5820-7. 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).
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