# Group theory

2017-07-30T16:42:37+03:00[Europe/Moscow] en true Symmetry group, Representation theory, Group representation, Multiplicative group of integers modulo n, Braid group, Kleinian group, Group (mathematics), Point group, Normal subgroup, Class function (algebra), Orbifold notation, List of character tables for chemically important 3D point groups, Coxeter group, N-group (category theory), Modular group, Word (group theory), Symmetry in quantum mechanics, Abelian group, Lorentz group, Irreducible representation, 2-group, Number, Unit (ring theory), Group of Lie type, Group cohomology, Projective representation, Picard modular group, Symmetric group, Discrete logarithm, Space group, Dihedral symmetry in three dimensions, Coxeter element, Coxeter notation, Schnorr group flashcards Group theory
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• Symmetry group
In abstract algebra, 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.
• 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.
• 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.
• Multiplicative group of integers modulo n
In modular arithmetic the set of congruence classes relatively prime to the modulus number, say n, form a group under multiplication called the multiplicative group of integers modulo n.
• Braid group
In mathematics, the braid group on n strands (denoted Bn), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (eg. under ambient isotopy), and whose group operation is composition of braids (see ).
• Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C).
• 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.
• Point group
In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.
• 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.
• Class function (algebra)
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G.
• Orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.
• List of character tables for chemically important 3D point groups
This lists the character tables for the more common molecular point groups used in the study of molecular symmetry.
• Coxeter group
In mathematics, a Coxeter group, named after H.
• N-group (category theory)
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra.
• 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.
• Word (group theory)
In group theory, a word is any written product of group elements and their inverses.
• Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics.
• 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 (the axiom of commutativity).
• Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (nongravitational) physical phenomena.
• Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper subrepresentation closed under the action of .
• 2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid.
• Number
A number is a mathematical object used to count, measure, and label.
• Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.
• Group of Lie type
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.
• Group cohomology
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.
• 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 PGL(V, F) = GL(V, F) / F∗, where GL(V, F) is the general linear group of invertible linear transformations of V over F and F∗ is the normal subgroup consisting of multiplications of vectors in V by nonzero elements of F (that is, scalar multiples of the identity; scalar transformations).
• Picard modular group
In mathematics, a Picard modular group, studied by Picard (), is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and J is a hermitian form on L of signature (2, 1).
• Symmetric group
In abstract algebra, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.
• Discrete logarithm
In mathematics, a discrete logarithm is an integer k solving the equation bk = g, where b and g are elements of a finite group.
• Space group
In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.
• 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 ).
• Coxeter element
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.
• Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation, with modifiers to indicate certain subgroups.
• Schnorr group
A Schnorr group, proposed by Claus P.