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113-432: Various physical systems, such as crystals and 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

226-634: A binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold: Satisfying properties (1) and (2) means that a pairing is a function with domain X . It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y . Functions which satisfy property (3) are said to be " onto Y " and are called surjections (or surjective functions ). Functions which satisfy property (4) are said to be " one-to-one functions " and are called injections (or injective functions ). With this terminology,

339-638: 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

452-930: A grain boundary . Like a grain boundary, a twin boundary has different crystal orientations on its two sides. But unlike a grain boundary, the orientations are not random, but related in a specific, mirror-image way. Mosaicity is a spread of crystal plane orientations. A mosaic crystal consists of smaller crystalline units that are somewhat misaligned with respect to each other. In general, solids can be held together by various types of chemical bonds , such as metallic bonds , ionic bonds , covalent bonds , van der Waals bonds , and others. None of these are necessarily crystalline or non-crystalline. However, there are some general trends as follows: Metals crystallize rapidly and are almost always polycrystalline, though there are exceptions like amorphous metal and single-crystal metals. The latter are grown synthetically, for example, fighter-jet turbines are typically made by first growing

565-515: A molten condition nor entirely in solution, but the high temperature and pressure conditions of metamorphism have acted on them by erasing their original structures and inducing recrystallization in the solid state. Other rock crystals have formed out of precipitation from fluids, commonly water, to form druses or quartz veins. Evaporites such as halite , gypsum and some limestones have been deposited from aqueous solution, mostly owing to evaporation in arid climates. Water-based ice in

678-619: A molten fluid, or by crystallization out of a solution. Some ionic compounds can be very hard, such as oxides like aluminium oxide found in many gemstones such as ruby and synthetic sapphire . Covalently bonded solids (sometimes called covalent network solids ) are typically formed from one or more non-metals, such as carbon or silicon and oxygen, and are often very hard, rigid, and brittle. These are also very common, notable examples being diamond and quartz respectively. Weak van der Waals forces also help hold together certain crystals, such as crystalline molecular solids , as well as

791-463: 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

904-603: 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

1017-416: A "crystal" is based on the microscopic arrangement of atoms inside it, called the crystal structure . A crystal is a solid where the atoms form a periodic arrangement. ( Quasicrystals are an exception, see below ). Not all solids are crystals. For example, when liquid water starts freezing, the phase change begins with small ice crystals that grow until they fuse, forming a polycrystalline structure. In

1130-621: A bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from a set to itself is also called a permutation , and the set of all permutations of a set forms its symmetric group . Some bijections with further properties have received specific names, which include automorphisms , isomorphisms , homeomorphisms , diffeomorphisms , permutation groups , and most geometric transformations . Galois correspondences are bijections between sets of mathematical objects of apparently very different nature. For

1243-417: A bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and

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1356-438: 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

1469-429: 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

1582-555: A fraction of a millimetre to several centimetres across, although exceptionally large crystals are occasionally found. As of 1999 , the world's largest known naturally occurring crystal is a crystal of beryl from Malakialina, Madagascar , 18 m (59 ft) long and 3.5 m (11 ft) in diameter, and weighing 380,000 kg (840,000 lb). Some crystals have formed by magmatic and metamorphic processes, giving origin to large masses of crystalline rock . The vast majority of igneous rocks are formed from molten magma and

1695-405: A function f : X → Y is bijective if and only if it satisfies the condition Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs

1808-453: A function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y . Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible . A function is invertible if and only if it is a bijection. Stated in concise mathematical notation,

1921-509: 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

2034-420: 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

2147-439: 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

2260-418: 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

2373-446: 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

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2486-407: 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

2599-608: A perfect, exactly repeating pattern. However, in reality, most crystalline materials have a variety of crystallographic defects : places where the crystal's pattern is interrupted. The types and structures of these defects may have a profound effect on the properties of the materials. A few examples of crystallographic defects include vacancy defects (an empty space where an atom should fit), interstitial defects (an extra atom squeezed in where it does not fit), and dislocations (see figure at right). Dislocations are especially important in materials science , because they help determine

2712-529: A plane perpendicular to the axis of rotation. Crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms , molecules , or ions ) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape , consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation

2825-451: 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

2938-460: 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

3051-434: A single crystal of titanium alloy, increasing its strength and melting point over polycrystalline titanium. A small piece of metal may naturally form into a single crystal, such as Type 2 telluric iron , but larger pieces generally do not unless extremely slow cooling occurs. For example, iron meteorites are often composed of single crystal, or many large crystals that may be several meters in size, due to very slow cooling in

3164-717: A single solid. Polycrystals include most metals , rocks, ceramics , and ice . A third category of solids is amorphous solids , where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass , wax , and many plastics . Despite the name, lead crystal, crystal glass , and related products are not crystals, but rather types of glass, i.e. amorphous solids. Crystals, or crystalline solids, are often used in pseudoscientific practices such as crystal therapy , and, along with gemstones , are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of

3277-421: 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,

3390-443: 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

3503-435: A wide range of properties. Polyamorphism is a similar phenomenon where the same atoms can exist in more than one amorphous solid form. Crystallization is the process of forming a crystalline structure from a fluid or from materials dissolved in a fluid. (More rarely, crystals may be deposited directly from gas; see: epitaxy and frost .) Crystallization is a complex and extensively-studied field, because depending on

Group theory - Misplaced Pages Continue

3616-417: 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

3729-423: 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

3842-435: 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

3955-445: Is a noncrystalline form. Polymorphs, despite having the same atoms, may have very different properties. For example, diamond is the hardest substance known, while graphite is so soft that it is used as a lubricant. Chocolate can form six different types of crystals, but only one has the suitable hardness and melting point for candy bars and confections. Polymorphism in steel is responsible for its ability to be heat treated , giving it

4068-530: 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

4181-702: 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

4294-490: 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 ,

4407-468: Is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f : A′ → B′ , where A′ is a subset of A and B′ is a subset of B . When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation . An example is the Möbius transformation simply defined on the complex plane, rather than its completion to

4520-411: 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

4633-675: Is bijective if and only if there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f , such that each of the two ways for composing the two functions produces an identity function : g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for each x {\displaystyle x} in X {\displaystyle X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for each y {\displaystyle y} in Y . {\displaystyle Y.} For example,

Group theory - Misplaced Pages Continue

4746-419: 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

4859-414: 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

4972-575: Is impossible for an ordinary periodic crystal (see crystallographic restriction theorem ). The International Union of Crystallography has redefined the term "crystal" to include both ordinary periodic crystals and quasicrystals ("any solid having an essentially discrete diffraction diagram" ). Quasicrystals, first discovered in 1982, are quite rare in practice. Only about 100 solids are known to form quasicrystals, compared to about 400,000 periodic crystals known in 2004. The 2011 Nobel Prize in Chemistry

5085-479: Is its visible external shape. This is determined by the crystal structure (which restricts the possible facet orientations), the specific crystal chemistry and bonding (which may favor some facet types over others), and the conditions under which the crystal formed. By volume and weight, the largest concentrations of crystals in the Earth are part of its solid bedrock . Crystals found in rocks typically range in size from

5198-582: Is known as crystallography . The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification . The word crystal derives from the Ancient Greek word κρύσταλλος ( krustallos ), meaning both " ice " and " rock crystal ", from κρύος ( kruos ), "icy cold, frost". Examples of large crystals include snowflakes , diamonds , and table salt . Most inorganic solids are not crystals but polycrystals , i.e. many microscopic crystals fused together into

5311-473: Is mechanically very strong, the sheets are rather loosely bound to each other. Therefore, the mechanical strength of the material is quite different depending on the direction of stress. Not all crystals have all of these properties. Conversely, these properties are not quite exclusive to crystals. They can appear in glasses or polycrystals that have been made anisotropic by working or stress —for example, stress-induced birefringence . Crystallography

5424-401: 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

5537-402: Is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function , i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup . Another way of defining the same notion is to say that a partial bijection from A to B

5650-400: 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

5763-424: Is the science of measuring the crystal structure (in other words, the atomic arrangement) of a crystal. One widely used crystallography technique is X-ray diffraction . Large numbers of known crystal structures are stored in crystallographic databases . Bijection A bijection , bijective function , or one-to-one correspondence between two mathematical sets is a function such that each element of

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5876-562: 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

5989-424: Is the type of impurities present in a corundum crystal. In semiconductors , a special type of impurity, called a dopant , drastically changes the crystal's electrical properties. Semiconductor devices , such as transistors , are made possible largely by putting different semiconductor dopants into different places, in specific patterns. Twinning is a phenomenon somewhere between a crystallographic defect and

6102-407: 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 ,

6215-490: 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

6328-503: 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

6441-400: 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

6554-816: 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

6667-447: The mechanical strength of materials . Another common type of crystallographic defect is an impurity , meaning that the "wrong" type of atom is present in a crystal. For example, a perfect crystal of diamond would only contain carbon atoms, but a real crystal might perhaps contain a few boron atoms as well. These boron impurities change the diamond's color to slightly blue. Likewise, the only difference between ruby and sapphire

6780-430: The multiplication by two defines a bijection from the integers to the even numbers , which has the division by two as its inverse function. A function is bijective if and only if it is both injective (or one-to-one )—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto )—meaning that each element of the codomain is mapped from at least one element of

6893-547: 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,

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7006-424: The 19th century. One of 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

7119-449: 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

7232-429: The air ( ice fog ) more often grow from a supersaturated gaseous-solution of water vapor and air, when the temperature of the air drops below its dew point , without passing through a liquid state. Another unusual property of water is that it expands rather than contracts when it crystallizes. Many living organisms are able to produce crystals grown from an aqueous solution , for example calcite and aragonite in

7345-486: 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

7458-510: The case of most molluscs or hydroxylapatite in the case of bones and teeth in vertebrates . The same group of atoms can often solidify in many different ways. Polymorphism is the ability of a solid to exist in more than one crystal form. For example, water ice is ordinarily found in the hexagonal form Ice I h , but can also exist as the cubic Ice I c , the rhombohedral ice II , and many other forms. The different polymorphs are usually called different phases . In addition,

7571-621: 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

7684-413: The category Grp of groups , the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms. The notion of one-to-one correspondence generalizes to partial functions , where they are called partial bijections , although partial bijections are only required to be injective. The reason for this relaxation

7797-404: The composition g ∘ f {\displaystyle g\,\circ \,f} of two functions is bijective, it only follows that f is injective and g is surjective . If X and Y are finite sets , then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory , this is taken as

7910-404: The conditions, a single fluid can solidify into many different possible forms. It can form a single crystal , perhaps with various possible phases , stoichiometries , impurities, defects , and habits . Or, it can form a polycrystal , with various possibilities for the size, arrangement, orientation, and phase of its grains. The final form of the solid is determined by the conditions under which

8023-409: The crystal can shrink or stretch it. Another is birefringence , where a double image appears when looking through a crystal. Moreover, various properties of a crystal, including electrical conductivity , electrical permittivity , and Young's modulus , may be different in different directions in a crystal. For example, graphite crystals consist of a stack of sheets, and although each individual sheet

8136-411: The crystal is one grain in a polycrystalline solid. The flat faces (also called facets ) of a euhedral crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal : they are planes of relatively low Miller index . This occurs because some surface orientations are more stable than others (lower surface energy ). As a crystal grows, new atoms attach easily to

8249-532: The crystals may form hexagons, such as ordinary water ice ). Crystals are commonly recognized, macroscopically, by their shape, consisting of flat faces with sharp angles. These shape characteristics are not necessary for a crystal—a crystal is scientifically defined by its microscopic atomic arrangement, not its macroscopic shape—but the characteristic macroscopic shape is often present and easy to see. Euhedral crystals are those that have obvious, well-formed flat faces. Anhedral crystals do not, usually because

8362-419: The definition of "same number of elements" ( equinumerosity ), and generalising this definition to infinite sets leads to the concept of cardinal number , a way to distinguish the various sizes of infinite sets. Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in

8475-606: The degree of crystallization depends primarily on the conditions under which they solidified. Such rocks as granite , which have cooled very slowly and under great pressures, have completely crystallized; but many kinds of lava were poured out at the surface and cooled very rapidly, and in this latter group a small amount of amorphous or glassy matter is common. Other crystalline rocks, the metamorphic rocks such as marbles , mica-schists and quartzites , are recrystallized. This means that they were at first fragmental rocks like limestone , shale and sandstone and have never been in

8588-412: 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

8701-432: The domain. The term one-to-one correspondence must not be confused with one-to-one function , which means injective but not necessarily surjective. The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...) , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists

8814-449: The eight faces of the octahedron belong to another crystallographic form reflecting a different symmetry of the isometric system. A crystallographic form is described by placing the Miller indices of one of its faces within brackets. For example, the octahedral form is written as {111}, and the other faces in the form are implied by the symmetry of the crystal. Forms may be closed, meaning that

8927-415: 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

9040-595: 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

9153-710: The final block of ice, each of the small crystals (called " crystallites " or "grains") is a true crystal with a periodic arrangement of atoms, but the whole polycrystal does not have a periodic arrangement of atoms, because the periodic pattern is broken at the grain boundaries . Most macroscopic inorganic solids are polycrystalline, including almost all metals , ceramics , ice , rocks , etc. Solids that are neither crystalline nor polycrystalline, such as glass , are called amorphous solids , also called glassy , vitreous, or noncrystalline. These have no periodic order, even microscopically. There are distinct differences between crystalline solids and amorphous solids: most notably,

9266-424: 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

9379-520: The fluid is being solidified, such as the chemistry of the fluid, the ambient pressure , the temperature , and the speed with which all these parameters are changing. Specific industrial techniques to produce large single crystals (called boules ) include the Czochralski process and the Bridgman technique . Other less exotic methods of crystallization may be used, depending on the physical properties of

9492-415: The form can completely enclose a volume of space, or open, meaning that it cannot. The cubic and octahedral forms are examples of closed forms. All the forms of the isometric system are closed, while all the forms of the monoclinic and triclinic crystal systems are open. A crystal's faces may all belong to the same closed form, or they may be a combination of multiple open or closed forms. A crystal's habit

9605-402: The form of snow , sea ice , and glaciers are common crystalline/polycrystalline structures on Earth and other planets. A single snowflake is a single crystal or a collection of crystals, while an ice cube is a polycrystal . Ice crystals may form from cooling liquid water below its freezing point, such as ice cubes or a frozen lake. Frost , snowflakes, or small ice crystals suspended in

9718-477: 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

9831-452: 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

9944-435: 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

10057-423: 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

10170-458: The interlayer bonding in graphite . Substances such as fats , lipids and wax form molecular bonds because the large molecules do not pack as tightly as atomic bonds. This leads to crystals that are much softer and more easily pulled apart or broken. Common examples include chocolates, candles, or viruses. Water ice and dry ice are examples of other materials with molecular bonding. Polymer materials generally will form crystalline regions, but

10283-471: The lengths of the molecules usually prevent complete crystallization—and sometimes polymers are completely amorphous. A quasicrystal consists of arrays of atoms that are ordered but not strictly periodic. They have many attributes in common with ordinary crystals, such as displaying a discrete pattern in x-ray diffraction , and the ability to form shapes with smooth, flat faces. Quasicrystals are most famous for their ability to show five-fold symmetry, which

10396-403: 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

10509-517: 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

10622-400: 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

10735-598: The player who will be batting in that position. The composition g ∘ f {\displaystyle g\,\circ \,f} of two bijections f : X → Y and g : Y → Z is a bijection, whose inverse is given by g ∘ f {\displaystyle g\,\circ \,f} is ( g ∘ f ) − 1 = ( f − 1 ) ∘ ( g − 1 ) {\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})} . Conversely, if

10848-413: The process of forming a glass does not release the latent heat of fusion , but forming a crystal does. A crystal structure (an arrangement of atoms in a crystal) is characterized by its unit cell , a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal. The symmetry of a crystal is constrained by

10961-432: The requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries (230 is commonly cited, but this treats chiral equivalents as separate entities), called crystallographic space groups . These are grouped into 7 crystal systems , such as cubic crystal system (where the crystals may form cubes or rectangular boxes, such as halite shown at right) or hexagonal crystal system (where

11074-423: The rougher and less stable parts of the surface, but less easily to the flat, stable surfaces. Therefore, the flat surfaces tend to grow larger and smoother, until the whole crystal surface consists of these plane surfaces. (See diagram on right.) One of the oldest techniques in the science of crystallography consists of measuring the three-dimensional orientations of the faces of a crystal, and using them to infer

11187-444: The same atoms may be able to form noncrystalline phases . For example, water can also form amorphous ice , while SiO 2 can form both fused silica (an amorphous glass) and quartz (a crystal). Likewise, if a substance can form crystals, it can also form polycrystals. For pure chemical elements, polymorphism is known as allotropy . For example, diamond and graphite are two crystalline forms of carbon , while amorphous carbon

11300-425: The same position in the list. In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion

11413-405: The second set (the codomain ) is the image of exactly one element of the first set (the domain ). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if and only if it is invertible ; that is, a function f : X → Y {\displaystyle f:X\to Y}

11526-479: The set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in

11639-474: 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

11752-731: The substance, including hydrothermal synthesis , sublimation , or simply solvent-based crystallization . Large single crystals can be created by geological processes. For example, selenite crystals in excess of 10  m are found in the Cave of the Crystals in Naica, Mexico. For more details on geological crystal formation, see above . Crystals can also be formed by biological processes, see above . Conversely, some organisms have special techniques to prevent crystallization from occurring, such as antifreeze proteins . An ideal crystal has every atom in

11865-671: 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

11978-656: 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

12091-429: The underlying crystal symmetry . A crystal's crystallographic forms are sets of possible faces of the crystal that are related by one of the symmetries of the crystal. For example, crystals of galena often take the shape of cubes, and the six faces of the cube belong to a crystallographic form that displays one of the symmetries of the isometric crystal system . Galena also sometimes crystallizes as octahedrons, and

12204-620: The vacuum of space. The slow cooling may allow the precipitation of a separate phase within the crystal lattice, which form at specific angles determined by the lattice, called Widmanstatten patterns . Ionic compounds typically form when a metal reacts with a non-metal, such as sodium with chlorine. These often form substances called salts, such as sodium chloride (table salt) or potassium nitrate ( saltpeter ), with crystals that are often brittle and cleave relatively easily. Ionic materials are usually crystalline or polycrystalline. In practice, large salt crystals can be created by solidification of

12317-523: 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

12430-400: Was awarded to Dan Shechtman for the discovery of quasicrystals. Crystals can have certain special electrical, optical, and mechanical properties that glass and polycrystals normally cannot. These properties are related to the anisotropy of the crystal, i.e. the lack of rotational symmetry in its atomic arrangement. One such property is the piezoelectric effect , where a voltage across

12543-433: 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

12656-428: Was that: The instructor was able to conclude that there were just as many seats as there were students, without having to count either set. A bijection f with domain X (indicated by f : X → Y in functional notation ) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general , yield

12769-561: 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

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