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84 relations: Abelian group, Abstract algebra, Ashoka Chakra, Automorphism, Cartesian coordinate system, Cayley table, Center (group theory), Chemistry, Circle, Commutative property, Complex conjugate, Complex number, Conjugacy class, Coordinate rotations and reflections, Coprime integers, Coxeter group, Cycle graph (algebra), Cyclic group, Dicyclic group, Dihedral group of order 6, Dihedral symmetry in three dimensions, Direct product of groups, Divisor, Equilateral triangle, Euclidean plane isometry, Euler's totient function, Examples of groups, Finite group, Flag of India, Function composition, Generalized dihedral group, Generating set of a group, Geometry, Graph theory, Group (mathematics), Group isomorphism, Group representation, Group theory, Holomorph (mathematics), Icosahedron, Identity element, Identity function, Infinite dihedral group, Infinite group, Inner automorphism, Integer, Inversion (discrete mathematics), Isomorphism, John Wiley & Sons, Klein four-group, ..., Linear map, List of small groups, Mathematics, Matrix (mathematics), Matrix multiplication, Modular arithmetic, Multiplicative group of integers modulo n, Normal subgroup, Octahedron, Order (group theory), Orthogonal group, Outer automorphism group, Point groups in two dimensions, Polygon, Presentation of a group, Quasidihedral group, Reflection (mathematics), Reflection symmetry, Regular polygon, Rotation, Rotation group SO(3), Rotation matrix, Rotational symmetry, Scalar multiplication, Semidirect product, Stop sign, Subgroup, Sylow theorems, Symmetric group, Symmetry, Symmetry group, T-group (mathematics), Tetrahedron, Wolfram Demonstrations Project. Expand index (34 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Ashoka Chakra
The Ashoka Chakra is a depiction of the dharmachakra; represented with 24 spokes.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Cartesian coordinate system
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
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Cayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.
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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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Chemistry
Chemistry is the scientific discipline involved with compounds composed of atoms, i.e. elements, and molecules, i.e. combinations of atoms: their composition, structure, properties, behavior and the changes they undergo during a reaction with other compounds.
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Circle
A circle is a simple closed shape.
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Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
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Complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.
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Coordinate rotations and reflections
In geometry, two-dimensional coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Cycle graph (algebra)
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Dicyclic group
In group theory, a dicyclic group (notation Dicn or Q4n) is a member of a class of non-abelian groups of order 4n (n > 1).
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Dihedral group of order 6
In mathematics, the smallest non-abelian group has 6 elements.
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Dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn (n ≥ 2).
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Direct product of groups
In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.
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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
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Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal.
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Euclidean plane isometry
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.
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Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.
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Examples of groups
Some elementary examples of groups in mathematics are given on Group (mathematics).
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Finite group
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
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Flag of India
The National Flag of India is a horizontal rectangular tricolour of India saffron, white and India green; with the Ashoka Chakra, a 24-spoke wheel, in navy blue at its centre.
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Function composition
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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Generalized dihedral group
In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups.
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Generating set of a group
In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Holomorph (mathematics)
In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains (copies of) the group and its automorphism group.
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Icosahedron
In geometry, an icosahedron is a polyhedron with 20 faces.
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Identity element
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, which leaves other elements unchanged when combined with them.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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Infinite dihedral group
In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.
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Infinite group
In group theory, an area of mathematics, an infinite group is a group, of which the underlying set contains an infinite number of elements.
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Inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Inversion (discrete mathematics)
In computer science and discrete mathematics a sequence has an inversion where two of its elements are out of their natural order.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Klein four-group
In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
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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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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Octahedron
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
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Outer automorphism group
In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.
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Point groups in two dimensions
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane.
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Polygon
In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Quasidihedral group
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2.
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Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
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Reflection symmetry
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.
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Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
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Rotation
A rotation is a circular movement of an object around a center (or point) of rotation.
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Rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.
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Rotation matrix
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.
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Rotational symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.
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Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).
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Semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.
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Stop sign
A stop sign is a traffic sign to notify drivers that they must come to a complete stop and make sure no other cars are coming before proceeding.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Sylow theorems
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.
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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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Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.
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Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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T-group (mathematics)
In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal.
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
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Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.
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222 symmetry, 322 symmetry, 422 symmetry, 522 symmetry, 622 symmetry, Dieder group, Dihedral Group, Dihedral Groups, Dihedral group D2, Dihedral group D3, Dihedral group D4, Dihedral group D5, Dihedral group D7, Dihedral symmetry, Dihedron group.
References
[1] https://en.wikipedia.org/wiki/Dihedral_group
