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For example, the fundamental group "counts" how many paths in the space are essentially different. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg—MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups.

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Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. Algebraic geometry likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.

They are both theoretically and practically intriguing. Toroidal embeddings have recently led to advances in algebraic geometry , in particular resolution of singularities. Algebraic number theory makes uses of groups for some important applications. For example, Euler's product formula ,. The failure of this statement for more general rings gives rise to class groups and regular primes , which feature in Kummer's treatment of Fermat's Last Theorem.

Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures , that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. 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. The presence of the periodicity in the circle of fifths yields applications of elementary group theory in musical set theory.

In physics , groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem , every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.


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In chemistry and materials science , groups are used to classify crystal structures , regular polyhedra, and the symmetries of molecules. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur.

In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration.

In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule. In chemistry , there are five important symmetry operations. The identity operation E consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation.

Rotation around an axis C n consists of rotating the molecule around a specific axis by a specific angle. Other symmetry operations are: reflection, inversion and improper rotation rotation followed by reflection. Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs , relating to the summing of an infinite number of probabilities to yield a meaningful solution. Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography.

Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher , may also be interpreted as a very easy group operation. Most cryptographic schemes use groups in some way.

In particular Diffie—Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group. Group theory has three main historical sources: number theory , the theory of algebraic equations , and geometry. The number-theoretic strand was begun by Leonhard Euler , and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.

Early results about permutation groups were obtained by Lagrange , Ruffini , and Abel in their quest for general solutions of polynomial equations of high degree. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein 's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois , in the s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups.

The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries such as euclidean , hyperbolic or projective geometry using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie , in , started using groups now called Lie groups attached to analytic problems.

Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory , and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups. From Wikipedia, the free encyclopedia.

This article covers advanced notions. For basic topics, see Group mathematics. For group theory in social sciences, see social group. Basic notions. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics.

Finite groups. Discrete groups Lattices. Topological and Lie groups. Algebraic groups. Linear algebraic group Reductive group Abelian variety Elliptic curve. Main article: Group mathematics. Main article: Finite group. Main article: Representation theory.

Main article: Lie theory. Main article: Geometric group theory. Main article: Symmetry group. Main article: Galois theory. Main article: Algebraic topology. Main article: Algebraic geometry.

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Main article: Algebraic number theory. Main article: Harmonic analysis. Main article: History of group theory. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.

Representation Theory and Expansion in Groups I - Avi Wigderson

Acta Mathematica. Inorganic Chemistry 5th ed. Freeman, W. Group theory at Wikipedia's sister projects. Categories : Group theory. Hidden categories: All articles with unsourced statements Articles with unsourced statements from June Articles with unsourced statements from December Wikipedia articles with GND identifiers Wikipedia articles with NDL identifiers. Namespaces Article Talk. Views Read Edit View history.

By using this site, you agree to the Terms of Use and Privacy Policy. Basic notions Subgroup Normal subgroup Quotient group Semi- direct product. List of group theory topics. Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic. Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier.

This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject.

A number of exercises are included. However, it can also be used as a reference for workers in all areas of mathematics and statistics. Recensie s From the reviews: The aim of the author is to give an introductory text on ordinary representation theory of finite groups which is accessible for advanced undergraduates already. And Steinberg manages this outstandingly well.

Kowol, Monatshefte fur Mathematik, Vol.

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The exercises provide more examples and further common results. It is the applications that Steinberg uses to motivate the subject that make this text both interesting and valuable. Overall, a very user-friendly text with many examples and copious details. Summing Up: Recommended. Zerger, Choice, Vol. This book is an introductory course and it could be used by mathematicians and students who would like to learn quickly about the representation theory and character theory of finite groups, and for non-algebraists, statisticians and physicists who use representation theory.

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Jamshid Moori, Mathematical Reviews, Issue j The required background as to this introductory course on group representations, is in the level of linear algebra, group theory and some ring theory. This is an impressive and useful text, and should be looked at by anybody with an interest in the subject. Toon meer Toon minder. Reviews Schrijf een review. Kies je bindwijze.