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108 relations: ADE classification, Algebra, Algebraic group, Alternating group, American Mathematical Society, Automorphism, Évariste Galois, Big O notation, Binary icosahedral group, Birational geometry, Block design, Buckminsterfullerene, Camille Jordan, Center (group theory), Central series, Characteristic (algebra), Classical group, Collinearity, Collineation, Compound of five tetrahedra, Congruence subgroup, Covering group, Covering groups of the alternating and symmetric groups, Covering space, Cremona group, Cross-ratio, Determinant, Diagonal matrix, Division ring, Empty set, Ernst Witt, Fano plane, Fiber bundle, Field (mathematics), Finite field, Gaussian integer, General linear group, Golden ratio, Graduate Studies in Mathematics, Graduate Texts in Mathematics, Group action, Group extension, Group homomorphism, Group theory, Homogeneous coordinates, Homography, Homotopy group, Hurwitz surface, Hurwitz's automorphisms theorem, Icosahedral symmetry, ..., Incidence matrix, Incidence structure, Issai Schur, Jean-Pierre Serre, Journal de Mathématiques Pures et Appliquées, Kernel (algebra), Klein quartic, Linear fractional transformation, Linear independence, List of finite simple groups, Macbeath surface, Mathematics, Mathieu group, Maximal compact subgroup, Möbius transformation, McKay graph, Modular curve, Modular group, Monster group, Morphism of algebraic varieties, Noam Elkies, Non-Desarguesian plane, Paley graph, Perfect group, Platonic solid, Principal homogeneous space, Projective geometry, Projective line, Projective orthogonal group, Projective representation, Projective space, Projective unitary group, PSL(2,7), Q-analog, Quadratic residue, Quasisimple group, Quotient group, Rational function, Representation theory, Root of unity, Schur multiplier, Semilinear map, Simple group, Simply connected space, Singleton (mathematics), SL2(R), Solvable group, Special linear group, Sporadic group, Springer Science+Business Media, Steiner system, Stereographic projection, Symmetric group, Symplectic group, Unit (ring theory), Vector space, Zassenhaus group, 2-transitive group. Expand index (58 more) »
ADE classification
In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams.
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Algebra
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Évariste Galois
Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician.
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Big O notation
Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.
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Binary icosahedral group
In mathematics, the binary icosahedral group 2I or is a certain nonabelian group of order 120.
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Birational geometry
In mathematics, birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets.
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Block design
In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowed at times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application.
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Buckminsterfullerene
Buckminsterfullerene is a type of fullerene with the formula C60.
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Camille Jordan
Marie Ennemond Camille Jordan (5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse.
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Center (group theory)
In abstract algebra, the center of a group,, is the set of elements that commute with every element of.
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Central series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Classical group
In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.
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Collinearity
In geometry, collinearity of a set of points is the property of their lying on a single line.
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Collineation
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear.
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Compound of five tetrahedra
The compound of five tetrahedra is one of the five regular polyhedral compounds.
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Congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.
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Covering group
In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p: G → H is a continuous group homomorphism.
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Covering groups of the alternating and symmetric groups
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups.
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Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
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Cremona group
In algebraic geometry, the Cremona group, introduced by, is the group of birational automorphisms of the n-dimensional projective space over a field k. It is denoted by Cr(Pn(k)) or Bir(Pn(k)) or Crn(k).
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Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.
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Fano plane
In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2.
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Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
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Graduate Studies in Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.
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Homography
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.
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Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
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Hurwitz surface
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface.
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Hurwitz's automorphisms theorem
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1).
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Icosahedral symmetry
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
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Incidence matrix
In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects.
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Incidence structure
In mathematics, an abstract system consisting of two types of objects and a single relationship between these types of objects is called an incidence structure.
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Issai Schur
Issai Schur (January 10, 1875 – January 10, 1941) was a Russian mathematician who worked in Germany for most of his life.
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Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
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Journal de Mathématiques Pures et Appliquées
The Journal de Mathématiques Pures et Appliquées is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874).
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Kernel (algebra)
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.
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Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed.
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Linear fractional transformation
In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.
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Linear independence
In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.
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List of finite simple groups
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
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Macbeath surface
In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathieu group
In the area of modern algebra known as group theory, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by.
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Maximal compact subgroup
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.
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Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.
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McKay graph
In mathematics, the McKay graph of a finite-dimensional representation V of a finite group G is a weighted quiver encoding the structure of the representation theory of G. Each node represents an irreducible representation of G. If \chi_i, \chi_j are irreducible representations of G then there is an arrow from \chi_i to \chi_j if and only if \chi_j is a constituent of the tensor product V\otimes\chi_i.
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Modular curve
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).
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Modular group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
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Monster group
In the area of modern algebra known as group theory, the Monster group M (also known as the Fischer–Griess Monster, or the Friendly Giant) is the largest sporadic simple group, having order The finite simple groups have been completely classified.
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Morphism of algebraic varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.
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Noam Elkies
Noam David Elkies (born August 25, 1966) is an American mathematician and professor of mathematics at Harvard University.
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Non-Desarguesian plane
In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' theorem, or in other words a plane that is not a Desarguesian plane.
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Paley graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue.
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Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial).
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Platonic solid
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.
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Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.
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Projective geometry
Projective geometry is a topic in mathematics.
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Projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.
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Projective orthogonal group
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V.
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Projective representation
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Projective unitary group
In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center,, embedded as scalars.
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PSL(2,7)
In mathematics, the projective special linear group PSL(2, 7) (isomorphic to GL(3, 2)) is a finite simple group that has important applications in algebra, geometry, and number theory.
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Q-analog
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.
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Quadratic residue
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
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Quasisimple group
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence such that E.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Rational function
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2(G, Z) of a group G. It was introduced by in his work on projective representations.
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Semilinear map
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K".
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Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.
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Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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Singleton (mathematics)
In mathematics, a singleton, also known as a unit set, is a set with exactly one element.
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SL2(R)
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.
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Solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
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Special linear group
In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.
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Sporadic group
In group theory, a discipline within mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Steiner system
The Fano plane is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ.
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Stereographic projection
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Zassenhaus group
In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.
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2-transitive group
A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points.
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Projective Special Linear group, Projective general linear group, Projective group, Projective linear transform, Projective special linear group.
References
[1] https://en.wikipedia.org/wiki/Projective_linear_group
