Quadratic Forms, Linear Algebraic Groups, and Cohomology: 18 (Developments in Mathematics)

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Amazon Drive Cloud storage from Amazon. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein , considers group actions on manifolds by homeomorphisms or diffeomorphisms.

Auel : Remarks on the Milnor conjecture over schemes

The groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold.

A typical way of specifying an abstract group is through a presentation by generators and relations ,. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism , as well as the classes of group with a given such property: Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups.

The new paradigm was of paramount importance for the development of mathematics: An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space , differentiable manifold , or algebraic variety. If the group operations m multiplication and i inversion ,.

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study.

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Topological groups form a natural domain for abstract harmonic analysis , whereas Lie groups frequently realized as transformation groups are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified.

There is a fruitful relation between infinite abstract groups and topological groups: A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups profinite groups: During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups , and other related groups.

One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups , which may be viewed as dealing with " continuous symmetry ", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

Saying that a group G acts on a set X means that every element of G defines a bijective map on the set X in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V via Schur's lemma.

Given a group G , representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: The totality of representations is governed by the group's characters.

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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.

Quadratic Forms, Linear Algebraic Groups, and Cohomology

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: 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. 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 and cryptography likewise uses group theory in many ways.

Group theory

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.

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They are both theoretically and practically intriguing.