Diagram techniques in group theory


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This course is a general introduction to group theory, with a strong orientation towards its use in Physics.

Lattice Diagrams of Subgroups Part 2

Symmetries are fundamental in Physics and are mathematically formalized by the concept of group. Therefore physicists need to know how to formulate a symmetry, how to use it and understand all the consequences of it. In this regard, the notion of representation is central, as it encapsulates the way physical quantities behave under the symmetry transformations.

After a brief review of some general aspects of groups, the course will focus on the study and use of their representations. A few concrete applications will show the efficiency of group methods in Physics. Specific learning outcomes of the teaching unit. Book Description Cambridge University Press, New Book.

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Paperback or Softback. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters.

For example, Fourier polynomials can be interpreted as the characters of U 1 , the group of complex numbers of absolute value 1 , acting on the L 2 -space of periodic functions. A Lie group is a group that is also a differentiable manifold , with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie , who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures , which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

They provide a natural framework for analysing the continuous symmetries of differential equations differential Galois theory , in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. Groups can be described in different ways. A more compact way of defining a group is by generators and relations , also called the presentation of a group.

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. 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.

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Treatment of Group Theory in Spectroscopy

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.

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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.

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.

Group Theory Webbook - blogs

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.


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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.

Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory
Diagram techniques in group theory Diagram techniques in group theory

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