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Mathematical group based upon a finite number of elements
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or
Finite_group
Theorem classifying finite simple groups
classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic
Classification of finite simple groups
Classification_of_finite_simple_groups
Type of mathematical group
In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G
Residually_finite_group
Commutative group (mathematics)
their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. An abelian group is a set A {\displaystyle A} , together
Abelian_group
Group that admits a formal description in terms of reflections
the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter
Coxeter_group
Group type in algebra
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
Finitely_generated_group
Group whose Cayley graph is an initially subamenable graph
group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such
Sofic_group
Representations of finite groups, particularly on vector spaces
permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves
Representation theory of finite groups
Representation_theory_of_finite_groups
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
List_of_finite_simple_groups
Mathematical group
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
Group_of_Lie_type
Topological group that is in a certain sense assembled from a system of finite groups
profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide
Profinite_group
Finite simple group type not classified as Lie, cyclic or alternating
classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the
Sporadic_group
Generalization of the discrete Fourier transform
Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Branch of mathematics that studies the properties of groups
and 2004, that culminated into a complete classification of finite simple groups. Group theory has three main historical sources: number theory, the
Group_theory
Commutative group where every element is the sum of elements from one finite subset
abstract algebra, an abelian group ( G , + ) {\displaystyle (G,+)} is called finitely generated if there exist finitely many elements x 1 , … , x s {\displaystyle
Finitely generated abelian group
Finitely_generated_abelian_group
Specification of a mathematical group by generators and relations
combinatorial group theory. A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said
Presentation_of_a_group
Type of mathematical object
several constructions. Finite direct products of group schemes have a canonical group scheme structure. Given an action of one group scheme on another by
Group_scheme
Mathematical property
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
Finiteness properties of groups
Finiteness_properties_of_groups
Group without normal subgroups other than the trivial group and itself
smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually
Simple_group
Type of group
mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter
Locally_finite_group
Mathematical group that can be generated as the set of powers of a single element
generator of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic
Cyclic_group
Abstract algebra concept
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
Generating_set_of_a_group
Cardinality of a mathematical group, or of the subgroup generated by an element
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
Order_(group_theory)
Set with associative invertible operation
the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple
Group_(mathematics)
Group with subnormal series where all factors are abelian
particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In particular, the quaternion group is a solvable group given by the group extension
Solvable_group
Type of mathematical group
is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K). Any finite group is linear, because it can
Linear_group
Mathematics book by John Conway
The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton
ATLAS_of_Finite_Groups
Linear representation in mathematics
particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional
Steinberg_representation
Sporadic simple group
41 · 47 · 59 · 71 ≈ 8.0802 × 1053 . The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite
Monster_group
Group in which the order of every element is a power of p
G. Every finite p-group is nilpotent. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer
P-group
Mathematical group
automorphism group of its fundamental group. For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic
Outer_automorphism_group
Group homomorphism into the general linear group over a vector space
are: Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory
Group_representation
Group of even permutations of a finite set
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
Alternating_group
Type of group in abstract algebra
of functions. In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols
Symmetric_group
The Hecke algebra of a finite group is the algebra spanned by the double cosets HgH of a subgroup H of a finite group G. It is a special case of a Hecke
Hecke algebra of a finite group
Hecke_algebra_of_a_finite_group
Group in which each element has finite order
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such
Torsion_group
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
Thin group (finite group theory)
Thin_group_(finite_group_theory)
Transformations induced by a mathematical group
of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL (
Group_action
Type of solvable group in mathematics
polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are
Polycyclic_group
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
N-group_(finite_group_theory)
Sporadic simple group
doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012 MathWorld: McLaughlin group Atlas of Finite Group Representations: McLaughlin group
McLaughlin_sporadic_group
Group for which a given group is a normal subgroup
by some. Since any finite group G {\displaystyle G} possesses a maximal normal subgroup N {\displaystyle N} with simple factor group G / ι ( N ) {\displaystyle
Group_extension
geometrically finite if it can be described in terms of geometrically finite groups. A convex polyhedron C in hyperbolic space is called geometrically finite if
Geometric_finiteness
On graphs with given symmetry groups
It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G {\displaystyle G} ,
Frucht's_theorem
Subgroup of a root system's isometry group
finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups,
Weyl_group
Set of finitely supported functions from a group to a ring
the group algebra of a finite group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra
Group_ring
Type of object in algebraic geometry
an orbifold, while still allowing mild stacky phenomena such as finite stabilizer groups. More precisely, a stack F {\displaystyle F} over schemes is Deligne–Mumford
Deligne–Mumford_stack
Sporadic simple group
Zbl 1203.20012 MathWorld: Conway Groups Atlas of Finite Group Representations: Co3 version 2 Atlas of Finite Group Representations: Co3 version 3
Conway_group_Co3
Every subgroup of a cyclic group is cyclic, and if finite, its order divides its parent's
In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of
Subgroups_of_cyclic_groups
248-dimensional exceptional simple Lie group
connected real Lie group forms of E8 are therefore not algebraic and admit no faithful finite-dimensional representations. Over finite fields, the Lang–Steinberg
E8_(mathematics)
Mathematical abelian group
product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German:
Klein_four-group
Any of certain special normal subgroups of a group
arbitrary groups. In this section G will denote a finite group, though some aspects generalize to locally finite groups and to profinite groups. For a prime
Core_(group_theory)
History of a branch of mathematics
the affine group of an affine space over a finite field of prime order. Groups similar to Galois groups are (today) called permutation groups. The theory
History_of_group_theory
Concept in mathematical group theory
of finite groups use characters of modular representations. Characters of irreducible representations encode many important properties of a group and
Character_theory
Sporadic simple group
projective special linear group of 3-dimensional space over the finite field with 4 elements (Dixon & Mortimer 1996, pp. 192–205). This group, sometimes called
Mathieu_group_M24
Group of symmetries of a regular polygon
examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry
Dihedral_group
Tools for studying groups based on techniques from algebraic topology
treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups. The group homology H ∗ ( G , k
Group_cohomology
Discrete group type in group theory
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space
Reflection_group
Mathematical concept
group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite
Nilpotent_group
Theorems that help decompose a finite group based on prime factors of its order
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician
Sylow_theorems
of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Concept in abstract algebra
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
Special group (finite group theory)
Special_group_(finite_group_theory)
If G is a finitely generated group with exponent n, is G necessarily finite?
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_problem
Group that is also a differentiable manifold with group operations that are smooth
continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic
Lie_group
Concept in mathematics
of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field
Reductive_group
Sporadic simple group
597–614, doi:10.1017/S0305004100074454, ISSN 0305-0041, MR 1362942 MathWorld: Harada–Norton Group Atlas of Finite Group Representations: Harada–Norton group
Harada–Norton_group
Unsolved problem in mathematics
Unsolved problem in mathematics Is every finite group the Galois group of a Galois extension of the rational numbers? More unsolved problems in mathematics
Inverse_Galois_problem
Generalization of Lie groups
finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group
Schur_orthogonality_relations
Concerns the decomposition of representations of a finite group into irreducible pieces
Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's
Maschke's_theorem
Finite simple group; sometimes classed as sporadic
In group theory, the Tits group 2F4(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order 17,971,200 = 211 · 33 · 52 · 13
Tits_group
Four finite groups derived from the Leech lattice
algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced
Conway_group
From an exceptional automorphism of a Dynkin diagram
In mathematics, a Ree group is a group of Lie type over a finite field. They are named after Rimhak Ree, who constructed them from an exceptional automorphism
Ree_group
Algebraic structure
a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field
Finite_field
permutation groups and the matrix groups. The upper bound on the order of G given by |G| ≤ 2N shows that G is finite. The black box groups were introduced
Black_box_group
Area of mathematics
representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Problem in finite group theory
of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G {\displaystyle G} is the algorithmic problem
Word_problem_for_groups
Sporadic simple group
ISBN 978-1-84800-987-5, Zbl 1203.20012 Wilson, R. A. ATLAS of Finite Group Representations. MathWorld: Fischer Groups Atlas of Finite Group Representations: Fi22
Fischer_group_Fi22
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Commutative group in which all nonzero elements have the same order
abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary
Elementary_abelian_group
Sporadic simple group
Zbl 1203.20012 MathWorld: Conway Groups Atlas of Finite Group Representations: Co1 version 2 Atlas of Finite Group Representations: Co1 version 3
Conway_group_Co1
Locally compact topological group with an invariant averaging operation
functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G
Amenable_group
Concept in mathematics
proved finite generation of the group, and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of
Mapping class group of a surface
Mapping_class_group_of_a_surface
In group theory, equivalence class under the relation of conjugation
{\displaystyle p} -subgroups of a finite group G {\displaystyle G} are conjugates of each other. Let G {\displaystyle G} be a group. Two elements a , b ∈ G {\displaystyle
Conjugacy_class
Concept in mathematics
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some
Frobenius_group
Concept in model theory
a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank
Stable_group
German mathematician (1882–1935)
all group actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of transformations G acting on a finite-dimensional
Emmy_Noether
Branch of mathematics that studies abstract algebraic structures
of finite groups that have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic groups over finite fields
Representation_theory
Theorem in group theory
In finite group theory, the Schreier conjecture asserts that the outer automorphism group of every finite simple group is solvable. It was proposed by
Schreier_conjecture
Concept in mathematics
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections:
Complex_reflection_group
Sporadic simple group
Galois group. The inverse Galois problem is solved for all other sporadic simple groups. Let F211 be the finite field with 211 elements. Its group of units
Mathieu_group_M23
Group whose operation is composition of permutations
elements. A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only if it is closed
Permutation_group
Pictorial representation of symmetry
algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin
Dynkin_diagram
Existence of group elements of prime order
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
Measure on group representations
is applied specifically in the context of the group G {\displaystyle G} being the finite symmetric group S n {\displaystyle S_{n}} – see below. It is named
Plancherel_measure
Symmetry group of a configuration in space
lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. A glide plane is a reflection
Space_group
FC-group A group is an FC-group if every conjugacy class of its elements has finite cardinality. finite group A finite group is a group of finite order, that
Glossary_of_group_theory
Second homology group of a group
multiplier M ( G ) {\displaystyle \operatorname {M} (G)} of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup
Schur_multiplier
Mathematical study of invariants under symmetries
determinant of X, when A is in SLn. Let G {\displaystyle G} be a group, and V {\displaystyle V} a finite-dimensional vector space over a field k {\displaystyle
Invariant_theory
permutable subgroup is normal. (The finiteness is used crucially in the proofs.) In summary, a subgroup H of a finite group G is permutable in G if and only
Quasinormal_subgroup
leading to the Iwahori–Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel
Hecke_algebra_of_a_pair
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