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109-459: In mathematical group theory , a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup . They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection. The simple C-groups were determined by Suzuki (1965) , and his classification is summarized by Gorenstein (1980 , 16.4). The classification of C-groups

218-412: A Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity

327-655: 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. Examples of the use of groups in physics include the Standard Model , gauge theory , the Lorentz group , and the Poincaré group . Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs , relating to

436-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

545-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

654-463: A presentation by generators and relations . The first class of groups to undergo a systematic study was permutation groups . Given any set X and a collection G of bijections of X into itself (known as permutations ) that is closed under compositions and inverses, G is a group acting on X . If X consists of n elements and G consists of all permutations, G is the symmetric group S n ; in general, any permutation group G

763-608: A torus . 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 , captures the fact that any integer decomposes in a unique way into primes . 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

872-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

981-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

1090-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

1199-439: A family of quotients which are finite p -groups of various orders, and properties of G translate into the properties of its finite quotients. 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 . As a consequence, the complete classification of finite simple groups

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1308-429: 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

1417-514: A geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph , whose vertices correspond to group elements and edges correspond to right multiplication in the group. 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

1526-420: 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: a representation of G on a vector space V is a group homomorphism : where GL ( V ) consists of the invertible linear transformations of V . In other words, to every group element g

1635-439: A group G acting in a reasonable manner on a metric space X , for example a compact manifold , then G is quasi-isometric (i.e. looks similar from a distance) to the space X . 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

1744-419: A group acts on the n -dimensional vector space K by linear transformations . This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G . Permutation groups and matrix groups are special cases of transformation groups : groups that act on a certain space X preserving its inherent structure. In

1853-446: 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 is associative. Frucht's theorem says that every group

1962-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

2071-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

2180-407: 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

2289-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

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2398-537: A paper of Burnside (1899) , which was forgotten for many years until rediscovered by Feit in 1970. The C-groups include as special cases the TI-groups (trivial intersection groups), that are groups in which any two Sylow 2-subgroups have trivial intersection. These were classified by Suzuki ( 1964 ), and the simple ones are of the form PSL 2 ( q ), PSU 3 ( q ), Sz( q ) for q a power of 2. Group theory Various physical systems, such as crystals and

2507-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

2616-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

2725-458: A plane perpendicular to the axis of rotation. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of

2834-451: A position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, methane and other tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in

2943-474: A problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in

3052-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

3161-460: 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

3270-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

3379-421: A specific angle. It is rotation through the angle 360°/ n , where n is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, n = 2 , since applying it twice produces the identity operation. In molecules with more than one rotation axis,

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3488-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

3597-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

3706-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

3815-444: A whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert , Emil Artin , Emmy Noether , and mathematicians of their school. 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

3924-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

4033-417: Is simple , i.e. does not admit any proper normal subgroups . This fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals . The next important class of groups is given by matrix groups , or linear groups . Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such

4142-423: Is a subgroup of the symmetric group of X . An early construction due to Cayley exhibited any group as a permutation group, acting on itself ( X = G ) by means of the left regular representation . In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5 , the alternating group A n

4251-563: Is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

4360-435: Is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G , the geometry and analysis pertaining to G yield important results about Γ . A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups ( profinite groups ): for example, a single p -adic analytic group G has

4469-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

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4578-530: Is a prominent application of this idea. 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

4687-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

4796-702: 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. They are identity operation ( E) , rotation operation or proper rotation ( C n ), reflection operation ( σ ), inversion ( i ) and rotation reflection operation or improper rotation ( S n ). The identity operation ( E ) consists of leaving

4905-490: 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 Poincaré conjecture , proved in 2002/2003 by Grigori Perelman ,

5014-412: Is assigned an automorphism ρ ( g ) such that ρ ( g ) ∘ ρ ( h ) = ρ ( gh ) for any h in G . This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group G : often, the group operation in G is abstractly given, but via ρ , it corresponds to the multiplication of matrices , which

5123-420: 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 12- periodicity in

5232-415: Is called a word . Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups . A fundamental theorem of this area

5341-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

5450-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

5559-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

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5668-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

5777-401: 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

5886-400: Is 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

5995-562: 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

6104-407: Is through a presentation by generators and relations , A significant source of abstract groups is given by the construction of a factor group , or quotient group , G / H , of a group G by a normal subgroup H . Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory . If a group G is a permutation group on a set X ,

6213-491: Is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts (see Maschke's theorem ). 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

6322-509: The L -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 . The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse , page 3. Lie groups represent

6431-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

6540-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

6649-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

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6758-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

6867-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

6976-400: The circle of fifths yields applications of elementary group theory in musical set theory . Transformational theory models musical transformations as elements of a mathematical group. 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

7085-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

7194-816: The group isomorphism problem , which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation ⟨ x , y ∣ x y x y x = e ⟩ , {\displaystyle \langle x,y\mid xyxyx=e\rangle ,} is isomorphic to the additive group Z of integers, although this may not be immediately apparent. (Writing z = x y {\displaystyle z=xy} , one has G ≅ ⟨ z , y ∣ z 3 = y ⟩ ≅ ⟨ z ⟩ . {\displaystyle G\cong \langle z,y\mid z^{3}=y\rangle \cong \langle z\rangle .} ) Geometric group theory attacks these problems from

7303-408: The hydrogen atom , and three of the four known fundamental forces in the universe, may be modelled by symmetry groups . Thus group theory and the closely related representation theory have many important applications in physics , chemistry , and materials science . Group theory is also central to public key cryptography . The early history of group theory dates from the 19th century. One of

7412-550: The presentation of a group. Given any set F of generators { g i } i ∈ I {\displaystyle \{g_{i}\}_{i\in I}} , the free group generated by F surjects onto the group G . The kernel of this map is called the subgroup of relations, generated by some subset D . The presentation is usually denoted by ⟨ F ∣ D ⟩ . {\displaystyle \langle F\mid D\rangle .} For example,

7521-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

7630-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

7739-449: The C n axis having the largest value of n is the highest order rotation axis or principal axis. For example in boron trifluoride (BF 3 ), the highest order of rotation axis is C 3 , so the principal axis of rotation is C 3 . In the reflection operation ( σ ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through

7848-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

7957-490: 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

8066-621: The case of permutation groups, X is a set; for matrix groups, X is a vector space . The concept of a transformation group is closely related with the concept of a symmetry group : transformation groups frequently consist of all transformations that preserve a certain structure. 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 . The groups themselves may be discrete or continuous . Most groups considered in

8175-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

8284-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

8393-412: 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. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h . A more compact way of defining a group is by generators and relations , also called

8502-415: 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

8611-595: The factor group G / H is no longer acting on X ; but the idea of an abstract group permits one not to worry about this discrepancy. 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: finite groups , periodic groups , simple groups , solvable groups , and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to

8720-424: 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 began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two isomorphic groups are considered as the same group. A typical way of specifying an abstract group

8829-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

8938-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

9047-431: The group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the Hodge conjecture (in certain cases).) The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric varieties are algebraic varieties acted on by

9156-453: The group operations m (multiplication) and i (inversion), are compatible with this structure, that is, they are continuous , smooth or regular (in the sense of algebraic geometry) maps, then G is a topological group , a Lie group , or an algebraic group . The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form

9265-435: The group presentation ⟨ a , b ∣ a b a − 1 b − 1 ⟩ {\displaystyle \langle a,b\mid aba^{-1}b^{-1}\rangle } describes a group which is isomorphic to Z × Z . {\displaystyle \mathbb {Z} \times \mathbb {Z} .} A string consisting of generator symbols and their inverses

9374-423: The horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation ( S n ) requires rotation of  360°/ n , followed by reflection through

9483-544: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

9592-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

9701-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

9810-403: 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. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis ( C n ) consists of rotating the molecule around a specific axis by

9919-400: The most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups . Group theory has three main historical sources: number theory , the theory of algebraic equations , and geometry . The number-theoretic strand

10028-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

10137-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

10246-523: The nascent theory of groups and field theory . 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 1830s, 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

10355-759: The oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem ,

10464-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

10573-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

10682-400: The plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σ h (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical ( σ v ) or dihedral ( σ d ). Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to

10791-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

10900-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

11009-474: 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

11118-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

11227-672: The summing of an infinite number of probabilities to yield a meaningful solution. In chemistry and materials science , point groups are used to classify regular polyhedra, and the symmetries of molecules , and space groups to classify crystal structures . The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality ), spectroscopic properties (particularly useful for Raman spectroscopy , infrared spectroscopy , circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals . Molecular symmetry

11336-606: 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 1884, 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

11445-523: Was achieved, meaning that all those simple groups from which all finite groups can be built are now known. 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

11554-433: 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. Évariste Galois coined the term "group" and established a connection, now known as Galois theory , between

11663-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

11772-561: Was unified starting around 1880. 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 . The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through

11881-609: Was used in Thompson's classification of N-groups . The finite non-abelian simple C-groups are The C-groups include as special cases the CIT-groups, that are groups in which the centralizer of any involution is a 2-group. These were classified by Suzuki ( 1961 , 1962 ), and the finite non-abelian simple ones consist of the finite non-abelian simple C-groups other than PSL 3 (2) and PSU 3 (2) for n ≥2. The ones whose Sylow 2-subgroups are elementary abelian were classified in

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