# Category: Group theory

Length function
In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homo
Multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: * the group under multiplication of the invertible elements of a field, ring, or other structur
No small subgroup
In mathematics, especially in topology, a topological group is said to have no small subgroup if there exists a neighborhood of the identity that contains no nontrivial subgroup of An abbreviation '"N
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorph
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedd
Opposite group
In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids, groups, r
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalar
Holomorph (mathematics)
In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorp
Schoof–Elkies–Atkin algorithm
The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in ellipt
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates grou
Demushkin group
In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a pro-p group G having a certain properties relating to duality in group cohomology. More precisely, G must be
One-relator group
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group
Power closed
In mathematics a p-group is called power closed if for every section of the product of powers is again a th power. Regular p-groups are an example of power closed groups. On the other hand, powerful p
Group structure and the axiom of choice
In mathematics a group is a set together with a binary operation on the set called multiplication that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form state
CN-group
In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of: are all groups of odd order solvable? Progress was made by showing that
Generalized polygon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and g
Perfect core
In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a
Finitely generated group
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elem
Artin transfer (group theory)
In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Origin
Diagonal subgroup
In the mathematical discipline of group theory, for a given group G, the diagonal subgroup of the n-fold direct product G  n is the subgroup This subgroup is isomorphic to G.
Buekenhout geometry
In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by .
Cycle graph (algebra)
In group theory, a subfield 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. A cycle i
Complement (group theory)
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that Equivalently, every element of G has a unique expressi
Virtually
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Giv
Automorphisms of the symmetric and alternating groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of stu
Multiplicative character
In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field, usually the field of complex n
Double coset
In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let G be a group, and let
Higman–Sims asymptotic formula
In finite group theory, the Higman–Sims asymptotic formula gives an asymptotic estimate on number of groups of prime power order.
Whitehead's algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group Fn. The algorithm is based on a classic 1936 paper of J.
Idempotent measure
In mathematics, an idempotent measure on a is a probability measure that equals its convolution with itself; in other words, an idempotent measure is an idempotent element in the topological semigroup
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 quan
Parity of a permutation
In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the o
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogo
Discrete logarithm
In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarith
Power automorphism
In mathematics, in the realm of group theory, a power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism
Direct sum of groups
In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can
PT-group
No description available.
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups. In some of the literature, 2-groups a
Van Kampen diagram
In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram ) is a planar diagram used to represent the fact that a particular word in t
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometr
Normal closure (group theory)
In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing
Higman group
In mathematics, the Higman group, introduced by Graham Higman, was the first example of an infinite finitely presented group with no non-trivial finite quotients. The quotient by the maximal proper no
List of finite spherical symmetry groups
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral,
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2,
Core (group theory)
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.
Free partially commutative group
No description available.
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 ∗. More precisely, H is a su
Nottingham group
In the mathematical field of infinite group theory, the Nottingham group is the group J(Fp) or N(Fp) consisting of formal power series t + a2t2+...with coefficients in Fp. The group multiplication is
Embedding problem
In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Ga
Lyndon–Hochschild–Serre spectral sequence
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating
Coset
In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right c
Symmetric set
In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.
Engel identity
The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the
Baby-step giant-step
In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite abelian group by Daniel Sha
Mennicke symbol
In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satisfying some identities found by . They were named by , who used them in their solution of th
Cannon–Thurston map
In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group
Schoof's algorithm
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the numb
Heisenberg group
In mathematics, the Heisenberg group , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be
Character table
In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of g
Measurable group
In mathematics, a measurable group is a special type of group in the intersection between group theory and measure theory. Measurable groups are used to study measures is an abstract setting and are o
Dedekind group
In group theory, a Dedekind group is a group G such that every subgroup of G is normal.All abelian groups are Dedekind groups.A non-abelian Dedekind group is called a Hamiltonian group. The most famil
Finiteness properties of groups
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is m
Component (group theory)
In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of
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. If and are two groups, then is an extension of by if there is a
Conjugation of isometries in Euclidean space
In a group, the conjugate by g of h is ghg−1.
Induced character
In mathematics, an induced character is the character of the representation V of a finite group G induced from a representation W of a subgroup H ≤ G. More generally, there is also a notion of of a cl
Outer automorphism group
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms.
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? Saharon Shelah proved that Whitehead'
Engel group
In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...[[x,y],y]
Schnorr group
A Schnorr group, proposed by Claus P. Schnorr, is a large prime-order subgroup of , the multiplicative group of integers modulo for some prime . To generate such a group, generate , , such that with ,
Moufang loop
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang. Smooth Moufan
Class function
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 other words
Idealizer
In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal. Such an idealizer is given by In ring theory, if A is an additive subgr
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.
Fibonacci group
In mathematics, for a natural number , the nth Fibonacci group, denoted or sometimes , is defined by n generators and n relations: * * * * * * . These groups were introduced by John Conway in 19
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation o
Generating set of a group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elemen
Cosocle
In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal norma
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 some of the group structure (the rest o
Moufang set
In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
Generalized dihedral group
In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral
Conjugacy problem
In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate element
Higman–Sims graph
In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pai
Word (group theory)
In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y,
Leinster group
In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. The Leinster groups are named after Tom Leinster, a mathematician at the Uni
Principalization (algebra)
In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractiona
Representation on coordinate rings
In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties. Let X be an affine algebraic variety over an algebraically closed field k o
Retract (group theory)
In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup.
Affine manifold
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset
Descendant tree (group theory)
In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order ,
Building (mathematics)
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projecti
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite cl
P-group generation algorithm
In mathematics, specifically group theory, finite groups of prime power order , for a fixed prime number and varying integer exponents , are briefly called finite p-groups. The p-group generation algo
Socle (mathematics)
In mathematics, the term socle has several related meanings.
Index calculus algorithm
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.Dedicated to the discrete logarithm in where is a prime, index calculus lead
Grigorchuk group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a fini
Hall–Janko graph
In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges. It is a rank 3 strongly re
Howson property
In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two
Principal ideal theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to th
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested
Pontryagin product
In mathematics, the Pontryagin product, introduced by Lev Pontryagin, is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontr
Burnside category
In category theory and homotopy theory the Burnside category of a finite group G is a category whose objects are finite G-sets and whose morphisms are (equivalence classes of) spans of G-equivariant m
Complex reflection group
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyper
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member
Polar point group
In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. The unmoved points will constitute a line, a plane, or all of spac
Bianchi group
In mathematics, a Bianchi group is a group of the form where d is a positive square-free integer. Here, PSL denotes the projective special linear group and is the ring of integers of the imaginary qua
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 le
Esquisse d'un Programme
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the
Category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group
Real tree
In mathematics, real trees (also called -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and
Class of groups
A class of groups is a set theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. This concept arose from t
Class automorphism
In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the a
Double group
The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's va
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. They can be realized via s
Von Neumann paradox
In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preser
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is
Landau's function
In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the larg
Spherical 3-manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the form where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and clos
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Coxeter complex
In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow t
Formation (group theory)
In group theory, a branch of mathematics, a formation is a class of groups closed under taking images and such that if G/M and G/N are in the formation then so is G/M∩N. introduced formations to unify
Measurable acting group
In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersecti
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. Every such group is a subgroup of the orthogo
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur in his work on projective representations.
Suzuki group
In the mathematical discipline known as group theory, the phrase Suzuki group refers to: * The Suzuki sporadic group, Suz or Sz is a sporadic simple group of order 213 · 37 · 52 · 7 · 11 · 13 = 448,3
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that
Invariant decomposition
The invariant decomposition is a decomposition of the elements of groups into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and clas
Representation ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classe
Burnside problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the
Group object
In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topol
Caesar cipher
In cryptography, a Caesar cipher, also known as Caesar's cipher, the shift cipher, Caesar's code or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of su
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G.The index is denoted or
Group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous
Hall's identity
In mathematics, Hall's identity may be: * The Hall–Witt identity * The Hall identity [ [x,y]2,z] = 0 for 2 by 2 matrices, showing that this is a polynomial identity ring * The Hall–Petresco identit
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials
Additive identity
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive
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. Here, may be any natural number or infin
Octacube (sculpture)
The Octacube is a large, stainless steel sculpture displayed in the mathematics department of Pennsylvania State University in State College, PA. The sculpture represents a mathematical object called
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite numb
Convergence group
In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian
Damm algorithm
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004.
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: It is a connected non-compact simple real Lie group of dimension 3 with applicatio
Polish group
No description available.
Remak decomposition
No description available.
Magnetic space group
In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron
Norm (group)
In mathematics, in the field of group theory, the norm of a group is the intersection of the normalizers of all its subgroups. This is also termed the Baer norm, after Reinhold Baer. The following fac
Cohomological dimension
In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topol
Bender's method
In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on
Maria Wonenburger
María Josefa Wonenburger Planells was a Galician mathematician who did research in the United States and Canada. She is known for her work on group theory. She was the first Spaniard to obtain a Fulbr
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 hermi
Group theory
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields,
Glossary of group theory
A group is a set together with an associative operation which admits an identity element and such that every element has an inverse. Throughout the article, we use to denote the identity element of a
Bicommutant
In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant
Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However,
Cremona group
In algebraic geometry, the Cremona group, introduced by Cremona , is the group of birational automorphisms of the -dimensional projective space over a field . It is denoted by or or . The Cremona grou
Ehlers group
In mathematical physics, the Ehlers group, named after Jürgen Ehlers, is a finite-dimensional transformation group of stationary vacuum spacetimes which maps solutions of Einstein's field equations to
Quasirandom group
In mathematics, a quasirandom group is a group that does not contain a large product-free subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake o
Wyckoff positions
In crystallography, a Wyckoff position is a point belonging to a set of points for which site symmetry groups are conjugate subgroups of the space group. Crystallography tables give the Wyckoff positi
Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other
Burnside ring
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation express
Hanna Neumann conjecture
In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was pose
Warfield group
In algebra, a Warfield group, studied by , is a summand of a simply presented abelian group.
Approximate group
In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correc
Group isomorphism problem
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups. The isomorphism problem was formulate
Homeomorphism group
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group oper
Centralizer and normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set of elements of G such that each member commutes with each element of S, or equiva
(B, N) pair
In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it
Center (group theory)
In abstract algebra, the center of a group, G, is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation, Z(G) = {z
IA automorphism
In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization. The abelianization of a group is its quotient by its commuta
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was
Quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is give
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two wor
Order (group theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length
Modular group
In mathematics, the modular group is the projective special linear group PSL(2, Z) of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular grou
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 nontri
Lovász conjecture
In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally Lás
Point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventio
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Con
Character group
In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special
Gain group
No description available.
Bol loop
In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in. A loop, L
Discrete symmetry
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by mul
Rank of a group
In the mathematical subject of group theory, the rank of a group G, denoted rank(G), can refer to the smallest cardinality of a generating set for G, that is If G is a finitely generated group, then t
P-compact group
In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was
Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky.
String group
In topology, a branch of mathematics, a string group is an infinite-dimensional group introduced by as a -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame b
Co-Hopfian group
In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.
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 (non-gravitational) physical phenomena. The
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element
Transfer (group theory)
In the mathematical field of group theory, the transfer defines, given a group G and a subgroup of finite index H, a group homomorphism from G to the abelianization of H. It can be used in conjunction
History of group theory
The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of alge
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. M
One-dimensional symmetry group
A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function f(x) for, say, the color at position x. The o
Additive group
An additive group is a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This
Subgroup method
The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word of an element. It doesn't always return the minimal word, but it can return optimal word
De Sitter invariant special relativity
In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter s
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 grou
List of group theory topics
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such a
Conjugacy class sum
In abstract algebra, a conjugacy class sum, or simply class sum, is a function defined for each conjugacy class of a finite group G as the sum of the elements in that conjugacy class. The class sums o
Hopfian object
In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object
Lattice of subgroups
In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion.In this lattice, the join of two subgroups
Absolutely convex set
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which
Orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensi
Bimonster group
In mathematics, the bimonster is a group that is the wreath product of the monster group M with Z2: The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-
Group analysis of differential equations
Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various transformations of independent and dependent
Real element
In group theory, a discipline within modern algebra, an element of a group is called a real element of if it belongs to the same conjugacy class as its inverse , that is, if there is a in with , where
Examples of groups
Some elementary examples of groups in mathematics are given on Group (mathematics).Further examples are listed here.
Frattini subgroup
In mathematics, particularly in group theory, the Frattini subgroup of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivia
Zappa–Szép product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in
Symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformatio
Small cancellation theory
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "smal
Induced representation
In group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given a representation of H, the induced representatio
Unit (ring theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative ide
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 an abstract group is a dihedral group Dihn
Hidden subgroup problem
The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring, discrete logarithm, graph isomorphism, and
Curie's Principle
Curie's Principle, or Curie's Symmetry Principle, is a maxim about cause and effect formulated by Pierre Curie in 1894: "the symmetries of the causes are to be found in theeffects". The idea was based
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. These tables are based on the group-theoretical treatment of the symmetry operations
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
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. There are four
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. Equivalently, a principal homogeneous
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
Fixed points of isometry groups in Euclidean space
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine spac
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is
Factor system
In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary
Hesse's principle of transfer
In geometry, Hesse's principle of transfer (German: Übertragungsprinzip) states that if the points of the projective line P1 are depicted by a rational normal curve in Pn, then the group of the projec
Acylindrically hyperbolic group
In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space
Resolvent (Galois theory)
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynom
Conjugacy class
In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called co
HNN extension
In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard Neumann, and Hanna Neu
Stability group
In mathematics, in the realm of group theory, the stability group of subnormal series is the group of automorphisms that act as identity on each quotient group. * v * t * e