An Introduction to the Theory of Groups

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The kernel of this map is called the subgroup of relations, generated by some subset D. Combinatorial group theory studies groups from the perspective of generators and relations. The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation.

The word problem asks whether two words are effectively the same group element.

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By relating the problem to Turing machines , one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem , which asks whether two groups given by different presentations are actually isomorphic. Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.

Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X , for example a compact manifold , then G is quasi-isometric i. Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example. The axioms of a group formalize the essential aspects of symmetry.

Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph.

So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms , and the symmetry group is the automorphism group of the object in question. Applications of group theory abound.

Almost all structures in abstract algebra are special cases of groups. Rings , for example, can be viewed as abelian groups corresponding to addition together with a second operation corresponding to multiplication.

Introduction to Sociology/Groups

Therefore, group theoretic arguments underlie large parts of the theory of those entities. Galois theory uses groups to describe the symmetries of the roots of a polynomial or more precisely the automorphisms of the algebras generated by these roots. The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory.

It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S 5 , the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can.

The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic topology is another domain which prominently associates groups to the objects the theory 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 the space is subjected to some deformation. 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. 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.

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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.

Representation Theory

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.

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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.

Modern theory of group actions and the special role of finite simple groups -

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.

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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.

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  4. This article covers advanced notions. For basic topics, see Group mathematics. For group theory in social sciences, see social group. These metrics are regularly updated to reflect usage leading up to the last few days. Citations are the number of other articles citing this article, calculated by Crossref and updated daily.

    Find more information about Crossref citation counts. The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric. Find more information on the Altmetric Attention Score and how the score is calculated. The main purpose of this article is to present the fundamentals and to define clearly the basic terms of group theory to prepare the reader for more advanced study of the subject.

    Cite this: J. Article Views Altmetric -. Citations 1.