Group theory

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.

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).

In mathematics, a discrete logarithm is an integer k solving the equation bk = g, where b and g are elements of a finite group.

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.

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.

A number is a mathematical object used to count, measure, and label.

In mathematics, a Coxeter group, named after H.

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.

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.

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.

In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C).

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 ).

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.

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.

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.

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.

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.

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter 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.

This lists the character tables for the more common molecular point groups used in the study of molecular symmetry.

In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid.

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.

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then gH = { gh : h an element of H } is the left coset of H in G with respect to g, andHg = { hg : h an element of H } is the right coset of H in G with respect to g.

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

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).

In group theory, a word is any written product of group elements and their inverses.

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.

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 .

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.

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.

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.

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).

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 ).

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.

A Schnorr group, proposed by Claus P.