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In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions
Basic_subgroup
Subset of a group that forms a group itself
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group
Subgroup
Subgroup invariant under conjugation
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation
Normal_subgroup
Commutative group (mathematics)
tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups
Abelian_group
Sporadic simple group
The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points respectively. M24 is the subgroup of S24
Mathieu_group_M24
Theorems that help decompose a finite group based on prime factors of its order
give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group
Sylow_theorems
Restriction on topological groups in mathematics
An abbreviation '"NSS"' is sometimes used. A basic example of a topological group with no small subgroup is the general linear group over the complex
No_small_subgroup
Subgroup of phyllosilicate minerals within the kaolinite-serpentine group
Serpentine subgroup (part of the kaolinite-serpentine group in the category of phyllosilicates) are greenish, brownish, or spotted minerals commonly found
Serpentine_subgroup
an observable subgroup of G if and only if the quotient variety G/K is a quasi-affine variety. Some basic facts about observable subgroups: Every normal
Observable_subgroup
Sporadic simple group
found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below. A Sylow 3-subgroup of J4 is a Heisenberg group: order 27,
Janko_group_J4
Algebraic variety with a group structure
algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic
Algebraic_group
Group of all devices on the same wireless network
segment. A service set is either a basic service set (BSS) or an extended service set (ESS). A basic service set is a subgroup, within a service set, of devices
Service_set_(802.11_network)
Subgroup of the group of invertible n×n matrices
In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)
Linear_algebraic_group
Group without normal subgroups other than the trivial group and itself
that characteristic as its subgroups. The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite
Simple_group
Type of group in abstract algebra
theorem states that every group G {\displaystyle G} is isomorphic to a subgroup of the symmetric group on (the underlying set of) G {\displaystyle G}
Symmetric_group
Sporadic simple group
mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson
Thompson_sporadic_group
come in basic sets of 24, eight of which commute with a given outside 3-transposition. The group Fi24 is not simple, but its derived subgroup has index
Fischer_group
Sporadic simple group
2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is
Monster_group
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
Hall_subgroup
Group that is also a differentiable manifold with group operations that are smooth
an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts. Let GL (
Lie_group
Group with subnormal series where all factors are abelian
solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof
Solvable_group
Sporadic simple group
in the OEIS). Wilson (1984) found the 15 conjugacy classes of maximal subgroups of Ru as follows: Griess (1982) Aschbacher, Michael; Smith, Stephen D
Rudvalis_group
Subgroup mapped to itself under every automorphism of the parent group
In group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation
Characteristic_subgroup
Mathematical theorem for algebraic structure of subgroups of free products
the Kurosh subgroup theorem for topological groups. In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural
Kurosh_subgroup_theorem
Sporadic simple group
3-cycles is normalized by the Fischer group Fi24, so He:2 is a subgroup of the derived subgroup Fi24' (the non-simple group Fi24 has 2 conjugacy classes of
Held_group
Sporadic simple group
subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi23 is a subgroup
Fischer_group_Fi23
Theorem classifying finite simple groups
2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the
Classification of finite simple groups
Classification_of_finite_simple_groups
Group of 𝑛 × 𝑛 invertible matrices
F)} or SL n ( F ) {\displaystyle \operatorname {SL} _{n}(F)} , is the subgroup of GL ( n , F ) {\displaystyle \operatorname {GL} (n,F)} consisting of
General_linear_group
Sporadic simple group
eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups of B which are listed in the table below. (Gorenstein 1993) Leon, Jeffrey
Baby_monster_group
Existence of group elements of prime order
order of any subgroup of a finite group G divides the order of G. In general, not every divisor of |G| arises as the order of a subgroup of G. Cauchy's
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
Sporadic simple group
O'Nan (1976) in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ ×Z/2nZ)
O'Nan_group
Theorem on the orders of subgroups
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 a divisor of |
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Monster and modular connection
quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of the Hecke congruence subgroup Γ0(p) in SL(2,R). They found that
Monstrous_moonshine
Disjoint, equal-size subsets of a group's underlying set
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
Coset
Group in which the order of every element is a power of p
Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime power pn that divides the order of G.
P-group
Finite simple group; sometimes classed as sporadic
by Jacques Tits (1964) who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group
Tits_group
Sporadic simple group
1986 Robert A. Wilson showed that J 1 {\displaystyle J_{1}} cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the
Janko_group_J1
Special types of subgroups encountered in group theory
S\subseteq G} fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers
Centralizer_and_normalizer
Group obtained by aggregating similar elements of a larger group
is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient
Quotient_group
Operation that combines groups
{\displaystyle G} and H {\displaystyle H} as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties
Free_product
Group that is a topological space with continuous group operations
H, which are open. If H is a subgroup of G, then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal
Topological_group
Mathematics concept
Nielsen–Schreier theorem: Every subgroup of a free group is free. Furthermore, if the free group F {\displaystyle F} has rank n and the subgroup H {\displaystyle H}
Free_group
Branch of mathematics that studies the properties of groups
G is the symmetric group Sn; in general, any permutation group G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited
Group_theory
Mathematical group that can be generated as the set of powers of a single element
group G, one can form the subgroup that consists of all its integer powers: ⟨g⟩ = { gk | k ∈ Z }, called the cyclic subgroup generated by g. The order
Cyclic_group
Sporadic simple group
centralized by the baby monster group, which therefore contains HN as a subgroup. Conway and Norton suggested in their 1979 paper that monstrous moonshine
Harada–Norton_group
Group of even permutations of a finite set
and denoted by An or Alt(n). For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements
Alternating_group
Type of mathematical group
G} has a central series of finite length. That is, a series of normal subgroups { 1 } = G 0 ◃ G 1 ◃ ⋯ ◃ G n = G {\displaystyle \{1\}=G_{0}\triangleleft
Nilpotent_group
Type of topological group
discrete if and only if its identity is isolated. A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace
Discrete_group
Operation in group theory
a subgroup H, and a normal subgroup N ◃ G {\displaystyle N\triangleleft G} , the following statements are equivalent: G is the product of subgroups, G
Semidirect_product
Discrete subgroup in a locally compact topological group
group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts
Lattice_(discrete_subgroup)
Concept in mathematics
reductive if the largest smooth connected unipotent normal subgroup of G is trivial. This normal subgroup is called the unipotent radical and is denoted Ru(G)
Reductive_group
Type of group in group theory
algebraic subgroup of G L n ( Q ) {\displaystyle \mathrm {GL} _{n}(\mathbb {Q} )} for some n {\displaystyle n} then we can define an arithmetic subgroup of G
Arithmetic_group
Sporadic simple group
over the field with 4 elements. This arises from the fact that Fi22 is a subgroup of 2E6(22). All the ordinary and modular character tables of Fi22 have
Fischer_group_Fi22
Cardinality of a mathematical group, or of the subgroup generated by an element
element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication
Order_(group_theory)
Transformations induced by a mathematical group
finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ( n , K ) {\displaystyle \operatorname
Group_action
Countries ranked by poverty rate
by its authorities. National estimates are based on population-weighted subgroup estimates from household surveys. Definitions of the poverty line vary
List of countries by percentage of population living in poverty
List_of_countries_by_percentage_of_population_living_in_poverty
Smallest normal group containing a set
{\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle G} containing S . {\displaystyle S.} Formally, if G
Normal_closure_(group_theory)
Sporadic simple group
Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice,
Suzuki_sporadic_group
Mathematical concept
direct product P as containing the original groups G and H as subgroups. These subgroups of P have the following three important properties: (Saying again
Direct_product_of_groups
Major subgroup of the Austronesian language family
The Malayo-Polynesian languages are a subgroup of the Austronesian languages, with approximately 385.5 million speakers. The Malayo-Polynesian languages
Malayo-Polynesian_languages
Branch of mathematics
Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas
Algebraic_topology
Algorithm for solving various problems in computational group theory
polynomial time. It was introduced by Sims in 1970, based on Schreier's subgroup lemma. The running time was subsequently improved by Donald Knuth in 1991
Schreier–Sims_algorithm
Number in {..., –2, –1, 0, 1, 2, ...}
also closed under subtraction. The integers form a ring which is the most basic one, in the following sense: for any ring, there is a unique ring homomorphism
Integer
Finite simple group type not classified as Lie, cyclic or alternating
is a subgroup S4×2F4(2)′ normalizing a 2C2 subgroup of the Baby monster B, giving rise to a subgroup 2·S4×2F4(2)′ normalizing a certain Q8 subgroup of the
Sporadic_group
Group of symmetries of a regular polygon
Dn is a subgroup of the symmetric group Sn for n ≥ 3. Since 2n > n! for n = 1 or n = 2, for these values, Dn is too large to be a subgroup. The inner
Dihedral_group
Type of mathematical object
type by letting A be a non-constant sheaf of abelian groups on S. For a subgroup scheme H of a group scheme G, the functor that takes an S-scheme T to G(T)/H(T)
Group_scheme
Mathematical group based upon a finite number of elements
classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. During the twentieth century, mathematicians investigated
Finite_group
Type of group in mathematics
connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of
Orthogonal_group
248-dimensional exceptional simple Lie group
group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group
E8_(mathematics)
Group of matrices with determinant 1
the n×n matrices with entries in R and determinant 1. This is a normal subgroup of the general linear group, since it is the kernel of the determinant
Special_linear_group
Concept in mathematics
has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion
Frobenius_group
Group of mathematical theorems
of f {\displaystyle f} is a normal subgroup of G {\displaystyle G} , The image of f {\displaystyle f} is a subgroup of H {\displaystyle H} , and The image
Isomorphism_theorems
Mathematical group
There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They
Group_of_Lie_type
Symmetry-based invariance to continuous group action
{\displaystyle G} is a group that acts on X {\displaystyle X} ; then a subgroup H ⊆ G {\displaystyle H\subseteq G} is a symmetry of f {\displaystyle f}
Continuous_symmetry
Group whose operation is composition of permutations
of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted
Permutation_group
Sporadic simple group
construction of the Lyons group, as an amalgam of its maximal 3-local subgroups. When the McLaughlin sporadic group was discovered, it was noticed that
Lyons_group
Set with associative invertible operation
notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties
Group_(mathematics)
Algebraic curve in mathematics
As for the groups constituting the torsion subgroup of E(Q), the following is known: the torsion subgroup of E(Q) is one of the 15 following groups (a
Elliptic_curve
Second homology group of a group
finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of M ( G ) {\displaystyle \operatorname
Schur_multiplier
Sporadic simple group
been implicitly found earlier by Coxeter (1958), who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with
Mathieu_group_M12
Orientation-preserving mapping class group of the torus
SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} is a subgroup of this group.) Similarly, PGL ( 2 , Z ) {\displaystyle \operatorname
Modular_group
Group of languages related through a common ancestor
relationships may be too remote to be detectable. Alternative explanations for some basic observed commonalities between languages include developmental theories
Language_family
Lie group homomorphism from the real numbers
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G {\displaystyle \varphi :\mathbb
One-parameter_group
Five sporadic simple groups
constructed in various ways. M12 has a simple subgroup of order 660, a maximal subgroup. That subgroup is isomorphic to the projective special linear
Mathieu_group
Concept in topology
compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal
Maximal_compact_subgroup
Sporadic simple group
constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions
Janko_group_J3
Mathematical function between groups that preserves multiplication structure
isomorphic to the quotient group G/ker h. The kernel of h is a normal subgroup of G. Assume u ∈ ker ( h ) {\displaystyle u\in \operatorname {ker} (h)}
Group_homomorphism
Four finite groups derived from the Leech lattice
any subgroup of Co0 that properly contains N; hence N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. N also is the subgroup in Co0
Conway_group
Sporadic simple group
sporadic groups and was discovered by Jack McLaughlin (1969) as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =
McLaughlin_sporadic_group
Sylow theorems Hall subgroup Wreath product Butterfly lemma Center of a group Centralizer and normalizer Characteristic subgroup Commutator Composition
List_of_group_theory_topics
Means of constructing a group from two subgroups
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
Direct_sum_of_groups
Construction in group theory
{\displaystyle H} on A Ω {\displaystyle A^{\Omega }} given above. The subgroup A Ω {\displaystyle A^{\Omega }} of A Ω ⋊ H {\displaystyle A^{\Omega }\rtimes
Wreath_product
for any pair g, h ∈ G. ascendant subgroup A subgroup H of a group G is ascendant if there is an ascending subgroup series starting from H and ending
Glossary_of_group_theory
Sporadic simple group
4-elements in the double cover 2.A100. The double cover 2.J2 occurs as a subgroup of the Conway group Co0. J2 is the only one of the 4 Janko groups that
Janko_group_J2
Periodic set of points
Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space. The requirements of minimum
Lattice_(group)
East Asian ethnic group
diverse Han subgroups, who display slight but discernible physical and physiological differences. Although genetically similar, Han Chinese subgroups exhibit
Han_Chinese
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
G^{X}} , given by the quotient of the automorphism group of LX by the subgroup of reflections- these are also known as the stabilizers of deep holes in
Umbral_moonshine
If G is a finitely generated group with exponent n, is G necessarily finite?
exponent n. By basic results of group theory, the intersection of two normal subgroups of finite index in any group is itself a normal subgroup of finite index
Burnside_problem
Representation of groups by permutations
that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group Sym(G) whose
Cayley's_theorem
Concept in "Pure Theory of Law" by Hans Kelsen
'Basic norm' (German: Grundnorm) is a concept in the Pure Theory of Law created by Hans Kelsen, a jurist and legal philosopher. Kelsen used this word
Basic_norm
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