Group (mathematics)

In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.

After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.

The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.

Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.

Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.

For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words

Geometric group theory

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act. Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects; this is done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is new, became a identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are substantial connections with complexity theory, mathematical logic, the study of Lie Groups and their discrete subgroups, dynamical systems, probability theory, K-theory, other areas of mathematics.

In the introduction to his book Topics in Geometric Group Theory, Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, reminds me of several things that Georges de Rham practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend". Geometric group theory grew out of combinatorial group theory that studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups. Combinatorial group theory as an area is subsumed by geometric group theory. Moreover, the term "geometric group theory" came to include studying discrete groups using probabilistic, measure-theoretic, arithmetic and other approaches that lie outside of the traditional combinatorial group theory arsenal.

In the first half of the 20th century, pioneering work of Max Dehn, Jakob Nielsen, Kurt Reidemeister and Otto Schreier, J. H. C. Whitehead, Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups. Other precursors of geometric group theory include Bass -- Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp, it studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie Groups Mostow rigidity theorem, the study of Kleinian groups, the progress achieved in low-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by William Thurston's Geometrization program.

The emergence of geometric group theory as a distinct area of mathematics is traced to the late 1980s and early 1990s. It was spurred by the 1987 monograph of Mikhail Gromov "Hyperbolic groups" that introduced the notion of a hyperbolic group, which captures the idea of a finitely generated group having large-scale negative curvature, by his subsequent monograph Asymptotic Invariants of Infinite Groups, that outlined Gromov's program of understanding discrete groups up to quasi-isometry; the work of Gromov had a transformative effect on the study of discrete groups and the phrase "geometric group theory" started appearing soon afterwards.. Notable themes and developments in geometric group theory in 1990s and 2000s include: Gromov's program to study quasi-isometric properties of groups. A influential broad theme in the area is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry.

This program involves: The study of properties. Examples of such properties of finitely generated groups include: the growth rate of a finitely generated group. Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space; this dire

Level set

In mathematics, a level set of a real-valued function f of n real variables is a set of the form L c =, that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. So a level curve is the set of all real-valued solutions of an equation in two variables x1 and x2; when n = 3, a level set is called a level surface, for higher values of n the level set is a level hypersurface. So a level surface is the set of all real-valued roots of an equation in three variables x1, x2 and x3, a level hypersurface is the set of all real-valued roots of an equation in n variables. A level set is a special case of a fiber. Level sets show up in many applications under different names. For example, an implicit curve is a level curve, considered independently of its neighbor curves, emphasizing that such a curve is defined by an implicit equation. Analogously, a level surface is sometimes called an isosurface.

The name isocontour is used, which means a contour of equal height. In various application areas, isocontours have received specific names, which indicate the nature of the values of the considered function, such as isobar, isogon, isochrone and indifference curve. Consider the 2-dimensional Euclidean distance: A level set L r of this function consists of those points that lie at a distance of r from the origin, otherwise known as a circle. For example, ∈ L 5, because d = 5. Geometrically, this means. More a sphere in a metric space with radius r centered at x ∈ M can be defined as the level set L r. A second example is the plot of Himmelblau's function shown in the figure to the right; each curve shown is a level curve of the function, they are spaced logarithmically: if a curve represents L x, the curve directly "within" represents L x / 10, the curve directly "outside" represents L 10 x. Theorem: If the function f is differentiable, the gradient of f at a point is either zero, or perpendicular to the level set of f at that point.

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, he decides to go in the direction where the slope is steepest; the other one is more cautious. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other. A consequence of this theorem is that if f is differentiable, a level set is a hypersurface and a manifold outside the critical points of f. At a critical point, a level set may be reduced to a point or may have a singularity such as a self-intersection point or a cusp. A set of the form L c − = is called a sublevel set of f. L c + = is called a superlevel set of f. Sublevel sets are important in minimization theory; the boundness of some non-empty sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum, by Weierstrass's