# Category: Finite groups

Icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the
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 symme
Thompson factorization
In mathematical finite group theory, a Thompson factorization, introduced by Thompson, is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of p-su
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)and in which composing any two of the three no
Uniqueness case
In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of c
P-stable group
In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Gorenstein and Walter in order to extend Thompson's uniqueness results in t
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
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective f
Regular p-group
In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian p-groups, but is general enough to include most "small" p-groups. Regular
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 modu
Brauer tree
In mathematics, in the theory of finite groups, a Brauer tree is a tree that encodes the characters of a block with cyclic defect group of a finite group. In fact, the trees encode the group algebra u
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
Group of GF(2)-type
In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type . As the name suggests, many of th
Dempwolff group
In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness
3D4
In mathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D4, depending on a cubic Galois extension of fields K ⊂
Puig subgroup
In mathematical finite group theory, the Puig subgroup, introduced by Puig, is a characteristic subgroup of a p-group analogous to the Thompson subgroup.
Black box group
In computational group theory, a black box group (black-box group) is a group G whose elements are encoded by bit strings of length N, and group operations are performed by an oracle (the "black box")
Sastry automorphism
In mathematics, a Sastry automorphism, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem of embedding Ree groups of type 2F4 into C
Diameter (group theory)
In the area of abstract algebra known as group theory, the diameter of a finite group is a measure of its complexity. Consider a finite group , and any set of generators S. Define to be the graph diam
Local analysis
In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the info
P-soluble group
No description available.
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. It is the autom
Cayley table
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a squar
Primary cyclic group
In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p.That is, it is a cyclic group of order pm, Cpm, for some prime number p, and
ATLAS of Finite Groups
The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, , Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational ass
Commuting probability
In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen
Thompson subgroup
In mathematical finite group theory, the Thompson subgroup of a finite p-group P refers to one of several characteristic subgroups of P. John G. Thompson originally defined to be the subgroup generate
Coherent set of characters
In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. Th
Dowling geometry
In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point. They are name
Space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid tra
Normal p-complement
In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product
Strongly embedded subgroup
In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H. The Be
3-transposition group
In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions, called the 3-transpositions, such that the product of any two involutions from the conju
Schur cover
No description available.
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
Octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same
Prime graph
In the mathematics of graph theory and finite groups, a prime graph is an undirected graph defined from a group. These graphs were introduced in a 1981 paper by J. S. Williams, credited to unpublished
Valentiner group
In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by Herman Valentiner in the form of an action of A6 on
Vertex of a representation
In mathematical finite group theory, the vertex of a representation of a finite group is a subgroup associated to it, that has a special representation called a source. Vertices and sources were intro
Carter subgroup
In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, a
Dihedral group of order 6
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3.
P-constrained group
In mathematics, a p-constrained group is a finite group resembling the centralizer of an element of prime order p in a group of Lie type over a finite field of characteristic p. They were introduced b
Characteristic 2 type
In finite group theory, a branch of mathematics, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristi
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangl
Ree group
In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, genera
Rubik's Cube group
The Rubik's Cube group is a group that represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set corresponds to a cube move, which is the effect of any sequence of rotati
Hall subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall.
Aschbacher block
In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbac
N-group (finite group theory)
In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified
Dicyclic group
In group theory, a dicyclic group (notation Dicn or Q4n, ⟨n,2,2⟩) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of
Iwasawa group
In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is per
Deligne–Lusztig theory
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by Pierre Deligne and Georg
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
Tate cohomology group
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced
Generalized Fitting subgroup
No description available.
Special abelian subgroup
In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup. Equival
Thin group (finite group theory)
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups of the 2-local subgroups are cyclic. Informally
Quasithin group
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characte
Finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a
2E6 (mathematics)
In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for t
A-group
In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and a
Special group (finite group theory)
In group theory, a discipline within abstract algebra, a special group is a finite group of prime power order that is either elementary abelian itself or of class 2 with its derived group, its center,
Hessian group
In mathematics, the Hessian group is a finite group of order 216, introduced by Jordan who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of t
Mitchell's group
In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by Mitchell. It has the structure 6.PSU4(F3).2. As a complex reflection group it ha
Isoclinism of groups
In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, althoug
Group of symplectic type
In mathematical finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic. According to , p.386), the p-groups of symplectic type were cl
Triple product property
In abstract algebra, the triple product property is an identity satisfied in some groups. Let be a non-trivial group. Three nonempty subsets are said to have the triple product property in if for all
Small subgroup confinement attack
In cryptography, a subgroup confinement attack, or small subgroup confinement attack, on a cryptographic method that operates in a large finite group is where an attacker attempts to compromise the me
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
Fitting subgroup
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Int
Fourier transform on finite groups
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
Suzuki groups
In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22n+1), 2B2(22n+1), Suz(22n+1), or G(22n+1), form an infinite family of groups of Lie type found by Suzuki, that a
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
Elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime,
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
Exceptional character
In mathematical finite group theory, an exceptional character of a group is a character related in a certain way to a character of a subgroup. They were introduced by , p. 663), based on ideas due to
In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by , is a finite group G together with a quadratic module, a faithful representation M on a vector space over the
Tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not nec
3-step group
In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step grou
Central product
In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the c
Trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial gro
Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: * in the study of finite groups, a Z-group is a finite group
Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating gro
Fibrifold
In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al., who
Monomial group
In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, in
List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
Signalizer functor
In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem
Thompson order formula
In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson , gives a formula for the order of a finite group in terms of the centralizers of involutions, exten
Pauli group
In physics and mathematics, the Pauli group on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix and all of the Pauli matrices , together with the products of these matric
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
Omega and agemo subgroup
In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe