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Abelian group in which every element can, in some sense, be divided by positive integers
In mathematics, specifically in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided
Divisible_group
Commutative group (mathematics)
a divisible group A is a subgroup of an abelian group G then A admits a direct complement: a subgroup C of G such that G=A⊕C. Thus divisible groups are
Abelian_group
In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic
Barsotti–Tate_group
Mathematical term in group theory
blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century. The Prüfer p-group may be identified
Prüfer_group
Shorthand way of determining whether a given number is divisible by a fixed divisor
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division
Divisibility_rule
Concept in philosophy and mathematics
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also
Infinite_divisibility
Type of mathematical group
groups, polycyclic groups, finitely generated linear groups, and fundamental groups of compact 3-manifolds. A divisible group is a group G where every g∈G
Residually_finite_group
Injective homomorphism
the categorical sense. For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are
Monomorphism
Integer that divides another integer
case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing
Divisor
Type of probability distribution
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary
Infinite divisibility (probability)
Infinite_divisibility_(probability)
Natural number
needed] A natural number is divisible by 3 if the sum of its digits in base 10 is also divisible by 3. This known as the divisibility rule of 3. Because of
3
Set with associative invertible operation
primality of p ensures that the usual product of two representatives is not divisible by p, and therefore that the modular product is nonzero. The identity
Group_(mathematics)
Sporadic group that is not a subquotient of the monster
dubbed the happy family. For example, the orders of J4 and the Lyons Group Ly are divisible by 37. Since 37 does not divide the order of the monster, these
Pariah_group
Mathematical object in abstract algebra
properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion
Injective_module
Group word game to teach mathematical division
buzz is a group word game for children to teach them about division. Players take turns to count incrementally, replacing any number divisible by three
Fizz_buzz
automorphism group Quotient group Examples of groups Abelian group Cyclic group Rank of an abelian group Dicyclic group Dihedral group Divisible group Finitely
List_of_group_theory_topics
Lie group of complex numbers of unit modulus; topologically a circle
topological group equipped with the discrete topology. The circle group is a divisible group. Its torsion subgroup is given by the set of all n-th roots of
Circle_group
geometry, the Serre–Tate theorem says that an abelian scheme and its p-divisible group have the same infinitesimal deformation theory. This was first proved
Serre–Tate_theorem
simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group. Simplicity: Simple for n ≥ 1. The derived group 2F4(2)′
List_of_finite_simple_groups
to check the condition on the group of rational numbers. Every divisible group or injective group (in particular the group of rational numbers Q {\displaystyle
Cotorsion_group
Mathematical group
Janko found the sporadic group J1. The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3. They have order 22(2n+1)(22(2n+1)
Group_of_Lie_type
Theorem classifying finite simple groups
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
sum of cyclic groups of order pn and infinite cyclic groups; B is a p-pure subgroup of A; The quotient group, A/B, is a p-divisible group. Conditions 1–3
Basic_subgroup
Existence of group elements of prime order
chosen freely, so X has |G|p−1 elements, which is divisible by p. Now from the fact that in a group if ab = e then ba = e, it follows that any cyclic
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
Mathematics concept
Hodge–Tate modules, using the work of Tate (1967) on p-divisible groups, and named them Mumford–Tate groups. The algebraic torus T used to describe Hodge structures
Mumford–Tate_group
Concept in mathematics
gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné
Formal_group_law
Subject area in mathematics
non-Archimedean local fields, the group K2(F) is the direct sum of a finite cyclic group of order m, say, and a divisible group K2(F)m. We have K2(Z) = Z/2
Algebraic_K-theory
Group in which the order of every element is a power of p
different groups of order at most 2000, 49487367289, or just over 99%, are 2-groups of order 1024. Every finite group whose order is divisible by p contains
P-group
mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such
Height_(abelian_group)
structures using the results of Tate (1967) on p-divisible groups. Suppose that G is the absolute Galois group of a p-adic field K. Then G has a canonical
Hodge–Tate_module
Algebraic structure
semigroup, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups is Krohn–Rhodes
Semigroup
Algebraic invariant
uniquely divisible group; K 2 M ( Q ) {\displaystyle K_{2}^{M}(\mathbb {Q} )} is the direct sum of the cyclic group of order 2 and cyclic groups of order
Milnor_K-theory
Mathematical object in category theory
isomorphism. In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily a divisible group. Assuming the axiom of choice,
Injective_object
Projective variety that is also an algebraic group
A/S, the group of n-torsion points forms a finite flat group scheme. The union of the pn-torsion points, for all n, forms a p-divisible group. Deformations
Abelian_variety
Type of mathematical object
de Fourier généralisée Shatz, Stephen S. (1986), "Group schemes, formal groups, and p-divisible groups", in Cornell, Gary; Silverman, Joseph H. (eds.),
Group_scheme
Mathematical group that can be generated as the set of powers of a single element
ordered by divisibility. Thus, since a prime number p has no nontrivial divisors, pZ is a maximal proper subgroup, and the quotient group Z/pZ is simple;
Cyclic_group
Mathematical group based upon a finite number of elements
simple group has order divisible by at least three distinct primes. The Feit–Thompson theorem, or odd order theorem, states that every finite group of odd
Finite_group
Category where every morphism is invertible; generalization of a group
Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function
Groupoid
Mathematical function between groups that preserves multiplication structure
its kernel consists of all integers that are divisible by 3. The set G≡{(ab01)|a>0,b∈𝐑} forms a group under matrix multiplication. For any complex number
Group_homomorphism
Magma obeying the Latin square property
algebraic structure that resembles a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and
Quasigroup
stack over the category of differentiable manifolds. p-divisible group See p-divisible group (roughly an analog of torsion points of an abelian variety)
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Infinite family of simple groups of Lie type
These simple groups are the only finite non-abelian ones with orders not divisible by 3. Suzuki (1960) originally constructed the Suzuki groups as subgroups
Suzuki_groups
Family of infinite discrete groups
are eligible for Garside methods based on the investigation of their divisibility relations, and are well understood: Artin–Tits monoids are cancellative
Artin–Tits_group
Group without normal subgroups other than the trivial group and itself
proper, non-identity subgroup, the group is not simple. Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes
Simple_group
Political term in the Commonwealth realms
politic (which never dies). The Crown and the sovereign are "conceptually divisible but legally indivisible [...] The office cannot exist without the office-holder"
The_Crown
Russian mathematician
(1950) On divisible groups with ascending central series. Doklady AN USSR 70, 965–968. Chernikov S.N. (1950) On a centralizer of divisible abelian normal
Sergei_Chernikov
Natural number
Although 1 meets the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1), by modern convention it is regarded as
1
Group in arithmetic geometry
pairing is an alternating form. The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true
Tate–Shafarevich_group
How spheres of various dimensions can wrap around each other
compact smooth spin 4-manifold is divisible by 16. Stable homotopy groups of spheres are used to describe the group Θn of h-cobordism classes of oriented
Homotopy_groups_of_spheres
Any of certain special normal subgroups of a group
Also, for a finite group, the p′,p-core is the intersection of the centralizers of the abelian chief factors whose order is divisible by p (all of which
Core_(group_theory)
Concept in abstract algebra
cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented
Subgroups_of_cyclic_groups
Austrian mathematician (born 1948)
2, pp. 153–182. MR 1411570 Rapoport, M (1996). Period spaces for p-divisible groups (PDF). Princeton, N.J: Princeton University Press. ISBN 978-0-691-02782-1
Michael_Rapoport
French mathematician (1924–1987)
one-dimensional formal groups also gave rise to the field of p-divisible groups. His major contributions were: The classification of p-adic Lie groups: every p-adic
Michel_Lazard
Task taxonomy
contributing to process losses within those groups. These tasks are divided into three categories: Component (or divisibility), Focus (quantity or quality), and
Steiner's_Taxonomy_of_Tasks
Number line and triangular tiling's symmetry mathematical structure
terms of core partitions (integer partitions in which no hook length is divisible by n) or bounded partitions (integer partitions in which no part is larger
Affine_symmetric_group
Algebra of formal sums
element y=x/n. In contrast, non-trivial free abelian groups are never divisible, because in a free abelian group the basis elements cannot be expressed as multiples
Free_abelian_group
Mathematical theory
of K) contained the same information. Both are equivalent to the p-divisible group associated to X, up to isogeny. Grothendieck conjectured that there
P-adic_Hodge_theory
difference. As Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } is a divisible group we just need to determine what happens on the cyclic subgroup generated
Generator_(category_theory)
Natural number
The sum of the even numbered groups is then subtracted from the sum of the odd numbered groups. The test number is divisible by 7, 11 or 13 iff the result
1001_(number)
example, a p-divisible group is an example of a fppf group sheaf (a group sheaf with respect to the fppf topology). automorphism group functor "Course
Group_functor
Branch of mathematics that studies algebraic structures
Divisible group Dedekind group, Hamiltonian group Examples of groups Trivial group Additive group Permutation group Symmetric group Alternating group
List of abstract algebra topics
List_of_abstract_algebra_topics
American mathematician
the International Congress of Mathematicians in Nice on p-divisible groups (Barsotti–Tate groups) that are important in algebraic geometry in prime characteristic
William_Messing
Type of field in mathematics
value group of F {\displaystyle F} is an extension by Z {\displaystyle \mathbb {Z} } (the value group of K {\displaystyle K} ) of a divisible group, with
P-adically_closed_field
Infinite sum that is considered independently from any notion of convergence
index set I associated to a non-zero coefficient. Finally if G is a divisible group and R is a real closed field, then R((G)) is a real closed field, and
Formal_power_series
American mathematician (1925–2019)
parametrization for certain p-adic elliptic curves and the p-divisible (Tate–Barsotti) groups. Many of his results were not immediately published and some
John_Tate_(mathematician)
Field in mathematics similar to the real numbers
ℝ[[G]] of formal power series on a totally ordered abelian divisible group G that is an η1 group of cardinality ℵ1 (Alling 1962). Ϝ however is not a complete
Real_closed_field
Natural number
three-digit base 10 polydivisible number, since 1 is divisible by 1, 10 is divisible by 2 and 102 is divisible by 3. This also shows that 102 is a Harshad number
102_(number)
Subgroup of an abelian group consisting of all elements of finite order
injective. Tensoring an abelian group A {\displaystyle A} with Q {\displaystyle \mathbb {Q} } (or any divisible group) kills torsion. That is, if T {\displaystyle
Torsion_subgroup
Branch of mathematics
deformations of abelian scheme A is controlled by deformations of the p-divisible group A [ p ∞ ] {\displaystyle A[p^{\infty }]} consisting of its p-power
Deformation_(mathematics)
How many times a number is divisible by 2
In mathematics, an even number (an integer that is divisible by 2) is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly
Singly_and_doubly_even
Branch of mathematical logic
of incomparables) Every countable abelian group is the direct sum of a divisible group and a reduced group. Determinacy for Σ0 1 ∧ Π0 1 games. Weaker
Reverse_mathematics
finite group is divisible by the order of every element. perfect core The perfect core of a group is its largest perfect subgroup. perfect group A perfect
Glossary_of_group_theory
Natural number
as the 12th century. A decimal integer is divisible by 9 if and only if the sum of its digits is divisible by 9. 9 is the only square number that is the
9
United States voting pattern
December 2001 cover story by David Brooks entitled, "One Nation, Slightly Divisible", illustrated.[original research?] Thus, red and blue became fixed in
Red_states_and_blue_states
If G is a finitely generated group with exponent n, is G necessarily finite?
of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite; together with the Novikov–Adian theorem,
Burnside_problem
Algebraic structure with an associative operation and an identity element
(one-element) monoid, which is also the trivial group. Every group is a monoid and every abelian group a commutative monoid. Any semigroup S may be turned
Monoid
Internationally accepted civil calendar
years is that every year divisible by four is a leap year, except for years that are divisible by 100 but not also divisible by 400. For example, 1800
Gregorian_calendar
and abelian groups a pure projective module amounts to a direct sum of cyclic groups. Divisible group Fuchs, L (1970), Infinite Abelian Groups, I, Pure and
Pure_subgroup
French mathematician (1944–2019)
needed] Among his first works was the classification of p-divisible groups (= Barsotti–Tate group) over the ring of integers of a local field and the field
Jean-Marc_Fontaine
Mathematical homomorphism
otherwise. Dieudonné module Demazure, Michel (1972), Lectures on p-divisible groups, Lecture Notes in Mathematics, vol. 302, Berlin, New York: Springer-Verlag
Verschiebung_operator
Natural number
3rd colossally abundant number, the 5th highly composite number, and is divisible by the numbers from 1 to 4, and 6, a comparatively large number of divisors
12_(number)
Means of constructing a group from two subgroups
\prod _{i\not =i_{0}}H_{i}} . If H {\displaystyle H} is a divisible subgroup of an abelian group G {\displaystyle G} then there exists another subgroup K
Direct_sum_of_groups
Natural number
ace of divisibility. The more divisible a number is ... the more useful it proves in certain situations. ... Is it because 60 is highly divisible that the
60_(number)
Unique numeric book identifier since 1970
digits (1st, 3rd, 5th, 7th, 9th, 11th, and 13th), the total will always be divisible by 10 (i.e., end in 0). A 10-digit ISBN is converted to a 13-digit ISBN
ISBN
Matrix of binary truth values
concepts, and then noting the induced concept lattice. Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted
Logical_matrix
Category whose objects are abelian groups and whose morphisms are group homomorphisms
is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator
Category_of_abelian_groups
Algebraic group theory concept
fact that for simply connected type C (I.e., symplectic groups) there are roots that are divisible (by 2) in the weight lattice; this gives rise to examples
Pseudo-reductive_group
Module over the non-commutative Dieudonné ring
that could be used to analyze p {\displaystyle p} -divisible groups. Galois actions on the group schemes transfer through the equivalences of categories
Dieudonné_module
Natural number
squares. Four is one of four all-Harshad numbers. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x=y2−z2
4
Multiple-winner electoral system
Cumulative voting (sometimes called the single divisible vote) is an election system where a voter casts multiple votes but can lump votes on a specific
Cumulative_voting
Theorems that help decompose a finite group based on prime factors of its order
for the negation of this statement. Theorem (1)—A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk. Proof Let
Sylow_theorems
Mathematics, group theory
{\displaystyle G} is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. The theorem was proved by William
Burnside's_theorem
Number of elements in a subset of a commutative group
isomorphic to Bn if and only if n is divisible by m. For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that K is
Rank_of_an_abelian_group
numbers ordered by divisibility. Its least element is 1, which divides all natural numbers, while its greatest element is 0, which is divisible by all positive
Division_lattice
the Brauer group of a local field K to the divisible group Q/Z. Every class in the Brauer group is represented by a class in the Brauer group of an unramified
Hasse_invariant_of_an_algebra
Decentralized digital cryptocurrency
capitalized and lowercase variants without distinction. One bitcoin is divisible to eight decimal places. Units for smaller amounts of bitcoin are the
Bitcoin
Algebra term
graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to 0 and that it is surjective. In the same paper, he made a conjecture
Witt_group
Matrix group
SLd(ℤ[1/p1,…,1/pr]). Let ΓS be an S-arithmetic group in an algebraic group 𝐆⊂GLd. If n is an integer not divisible by any prime in S, then all primes pi
Congruence_subgroup
Polynomial with indeterminate coefficients
is not divisible by eight. The quaternion group Q8. Heisenberg groups Hp3 for any odd prime p. The alternating group A4. The alternating group A5. Reflection
Generic_polynomial
French mathematician
Fargues' work mainly centers around the study of Shimura varieties, p-divisible groups and their moduli spaces, and p-adic Hodge theory. One of his most significant
Laurent_Fargues
travel, tourism, insurance
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