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Encyclopedia > Group theory

Group theory is that branch of mathematics concerned with the study of groups. These are sets with a rule, or operation, that allows any two elements to be combined to form a third element. The rule must have certain properties of familiar operations like addition and multiplication. (See precise definition below.) Group theory is used throughout mathematics and has several applications in physics and chemistry. Groups can be finite or infinite. A classification of finite simple groups, completed in 1983, is one of the major achievements of mathematics in the 20th century. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This picture illustrates how the hours in a clock form a group. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... In mathematics, multiplication is an elementary arithmetic operation. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Chemistry - the study of atoms, made of nuclei (conglomeration of center particles) and electrons (outer particles), and the structures they form. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

## Applications of group theory GA_googleFillSlot("encyclopedia_square");

Some important applications of group theory include:

• Groups are often used to capture the internal symmetry of other structures. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group. Also see automorphism group.
• Galois theory, which is the historical origin of the group concept, uses groups to describe the symmetries of the equations satisfied by the roots of a polynomial. The solvable groups are so-named because of their prominent role in this theory Galois theory was originally used to prove that polynomials of the fifth degree and higher cannot, in general, be solved in closed form, the way polynomials of lower degree can.
• Abelian groups, which add the commutative property a * b = b * a, underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.
• In algebraic topology, groups are used to describe 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. Examples include the fundamental group, homology groups and cohomology groups. The name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.
• An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. 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: Standard Model, Gauge theory, Lorentz group, Poincaré group

In mathematics, an invariant is something that does not change under a set of transformations. ... The symmetry group of an object (e. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... Burnsides lemma, sometimes also called Burnsides counting theorem, PÃ³lyas formula, the Cauchy-Frobenius lemma or the Orbit-Counting Theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics and mathematics, the PoincarÃ© group is the group of isometries of Minkowski spacetime. ... Chemistry - the study of atoms, made of nuclei (conglomeration of center particles) and electrons (outer particles), and the structures they form. ... Molecular symmetry in chemistry describes symmetry in molecules and the classification of molecules in groups based on symmetry. ... The polarity of an object is, in general, its physical alignment of atoms. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... This article or section does not adequately cite its references or sources. ... Infrared spectroscopy (IR Spectroscopy) is the subset of spectroscopy that deals with the IR region of the EM spectrum. ... This article or section does not adequately cite its references or sources. ... Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ...

## History

There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.[1] Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ... Galois at the age of fifteen from the pencil of a classmate. ... Field theory is a branch of mathematics which studies the properties of fields. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ...

An early source occurs in the problem of forming an mth-degree equation having as its roots m of the roots of a given nth-degree equation (m < n). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.[1] Johann van Waveren Hudde (April 23, 1628 - April 15, 1704) was a mathematician. ... Nicolas Saunderson (1682â€“April 9, 1739) was an English scientist and mathematician. ... Edward Waring (1736 - August 15, 1798) was British mathematician who was born in Old Heath (near Shrewsbury) Shropshire England and died in Pontesbury Shropshire England He was Lucasian professor of mathematics at Cambridge University from 1760 until his death. ...

A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.[1] This article is about permutation, a mathematical concept. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Alexandre-ThÃ©ophile Vandermonde (28 February 1735 -1 January 1796) was a French musician and chemist who worked with Bezout and Lavoisier; his name is now principally associated with determinant theory in mathematics. ...

Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.[1] Paolo Ruffini (Valentano, 1765 â€“ Modena, 1822) was an Italian mathematician and philosopher. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... In mathematics, the word transitive admits at least two distinct meanings: A group G acts transitively on a set S if for any x, y &#8712; S, there is some g &#8712; G such that gx = y. ... In mathematics, a permutation group G acting on a set X is primitive if G preserves no nontrivial partition of X. In the other case, G is imprimitive. ...

Galois found that if $r_1, r_2, ldots, r_n$ are the n roots of an equation, there is always a group of permutations of the r's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).[1] Galois at the age of fifteen from the pencil of a classmate. ... In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...

Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.[1] Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... Joseph Alfred Serret (August 30, 1819 - March 2, 1885) was a french mathematician who was born in Paris France and died in Versailles France. ... Marie Ennemond Camille Jordan (January 5, 1838 &#8211; January 22, 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours danalyse. ... Alternative meaning: Nineteenth Century (periodical) (18th century &#8212; 19th century &#8212; 20th century &#8212; more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Joseph Louis FranÃ§ois Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics. ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Emile Léonard Mathieu (May 15, 1835 - October 19, 1890) was a French mathematician. ...

It was Walther von Dyck who, in 1882, gave the modern definition of a group. Walther Franz Anton von Dyck (December 6, 1856 - November 5, 1934) was a German mathematician. ...

The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... Wilhelm Karl Joseph Killing (1847 May 10 &#8211; 1923 February 11) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. ... Eduard Study (23 March 1862 - 6 Jan 1930) was a 19th-century German mathematician known for work on invariant theory of ternary forms (1889). ... Issai Schur (January 10, 1875 in Mogilyov - January 10, 1941 in Tel Aviv) was a mathematician who worked in Germany for most of his life. ... Ã‰lie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Charles Ã‰mile Picard (July 24, 1856 - December 11, 1941) was a leading French mathematician. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ...

The classification of finite simple groups is a vast body of work from the mid 20th century, which is thought to classify all the finite simple groups. The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;1999... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...

Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others. Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... Amalie Emmy Noether [1] (March 23, 1882 â€“ April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... Peter Ludwig Mejdell Sylow (12 December 1832 &#8211;7 September 1918) was a Norwegian mathematician, who proved foundational results in group theory. ...

## Group theory concepts

### Definition of a group

Main article: Group (mathematics)

A group (G, *) is a set G with a binary operation * : G × GG (one that assigns each ordered pair (a,b) in G an element in G denoted by a*b (i.e., satisfies closure)) that satisfies the following 3 axioms: This picture illustrates how the hours in a clock form a group. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... Look up element in Wiktionary, the free dictionary. ... For the algebra software named Axiom, see Axiom computer algebra system. ...

• Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
• Identity element: There is an element e in G such that for all a in G, e * a = a * e = a.
• Inverse element: For each a in G, there is an element b in G such that a * b = b * a = e, where e is an identity element.

In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

### Subgroups

A set H is a subgroup of a group G if it is a subset of G and is a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

A subgroup H is a normal subgroup of G if for all h in H and g in G, ghg-1 is also in H. An alternative (but equivalent) definition is that a subgroup is normal if its left and right cosets coincide. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also sometimes called a factor group). In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

## Special classes of groups

A group is abelian (or commutative) if the operation is commutative (that is, for all a, b in G, a * b = b * a). A non-abelian group is a group that is not abelian. The term "abelian" is named after the mathematician Niels Abel. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Niels Henrik Abel (August 5, 1802&#8211;April 6, 1829), Norwegian mathematician, was born in Finnøy. ...

A cyclic group is a group that is generated by a single element. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

A simple group is a group that has no nontrivial normal subgroups. In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...

A solvable group , or soluble group, is a group that has a normal series whose quotient groups are all abelian. The fact that S5, the symmetric group in 5 elements, is not solvable proves that some quintic polynomials cannot be solved by radicals. In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ...

A free group is a group whose elements can be written uniquely as products (or strings) of elements from some subset of the group. Every group is the homomorphic image of some free group. The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many...

## Operations involving groups

A homomorphism is a map between two groups that preserves the structure imposed by the operator. If the map is bijective, then it is an isomorphism. An isomorphism from a group to itself is an automorphism. The set of all automorphisms of a group is a group called the automorphism group. The kernel of a homomorphism is a normal subgroup of the group. In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...

A group action is a map involving a group and a set, where each element in the group defines a bijective map on a set. Group actions are used to prove the Sylow theorems and to prove that the center of a p-group is nontrivial. In mathematics, a symmetry group describes all symmetries of objects. ... The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the... In abstract algebra, the centre of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z âˆˆ G | gz = zg for all g âˆˆ G} Note that Z(G) is a subgroup of G â€” if x and... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ...

## Some useful theorems

• Some basic results in elementary group theory
• Lagrange's theorem: if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G.
• Cayley's Theorem: every group G is isomorphic to a subgroup of the symmetric group on G.
• Sylow theorems: perhaps the most useful of the group theorems. Among them, that if pn (and p prime) divides the order of a finite group G, then there exists a subgroup of order pn.
• The Butterfly lemma is a technical result on the lattice of subgroups of a group.
• The Fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
• Jordan-Hölder theorem: any two composition series of a given group are equivalent.
• Krull-Schmidt theorem: a group G, subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite product of indecomposable subgroups.
• Burnside's lemma: the number of orbits of a group action on a set equals the average number of points fixed by each element of the group.

In mathematics, a group (G,*) is usually defined as: G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): A1. ... Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ... In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a... The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the... In mathematics, the butterfly lemma or Zassenhaus lemma is a technical result on the lattice of subgroups of a group. ... The lattice of subgroups of the dihedral group Dih4, represented as groups of rotations and reflections of a plane figure. ... In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. ... In mathematics, a composition series of a group G is a normal series such that each Hi is a maximal normal subgroup of Hi+1. ... The Krull-Schmidt theorem states that a group , subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite product of indecomposable subgroups. ... Burnsides lemma, sometimes also called Burnsides counting theorem, PÃ³lyas formula, the Cauchy-Frobenius lemma or the Orbit-Counting Theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ... In mathematics, a symmetry group describes all symmetries of objects. ...

## Miscellany

James Newman summarized group theory as follows: James Roy Newman was a mathematician and mathematical historian. ...

The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.

One application of group theory is in musical set theory. Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ...

In Philosophy, Ernst Cassirer related the theory of group to the theory of perception as described by Gestalt Psychology; Perceptual Constancy is taken to be analogous to the invariants of group theory. The philosopher Socrates about to take poison hemlock as ordered by the court. ... Ernst Cassirer (July 28, 1874 - April 13, 1945) was a German philosopher. ... This does not adequately cite its references or sources. ... Subjective constancy or perceptual constancy is the perception of an object or quality as constant under changing conditions. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share...

## References

1. ^ a b c d e f Smith, D. E., History of Modern Mathematics, Project Gutenberg, 1906.
• Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8.  A standard modern reference.
• Scott, W. R. [1964] (1987). Group Theory. New York: Dover. ISBN 0-486-65377-3.  An inexpensive and fairly readable textbook (somewhat outdated in emphasis, style, and notation).
• Livio, M. (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. ISBN 0-7432-5820-7.  Pleasant to read book that explains the importance of group theory and how its symmetries lead to parallel symmetries in the world of physics and other sciences. Really helps congeal the importance of group theory as a practical tool).

Project Gutenberg, abbreviated as PG, is a volunteer effort to digitize, archive, and distribute cultural works. ...

Results from FactBites:

 20: Group Theory and Generalizations (2774 words) Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way. Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology. Nielsen's theorem: subgroups of free groups are free.
 Group theory - Wikipedia, the free encyclopedia (1367 words) Group theory is that branch of mathematics concerned with the study of groups. A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. Group theory is also very important to the field of chemistry, where it is used to assign symmetries to molecules.
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