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Group whose subgroups are all normal
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
Dedekind_group
Algebra with unique prime factorization
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into
Dedekind_domain
In number theory, measure of non-unique factorization
example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. Ideal class groups (or, rather, what
Ideal_class_group
German mathematician (1831–1916)
Richard Dedekind Dedekind cut Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta
Richard_Dedekind
Sporadic simple group
^{2}+1240002q^{3}+10698752q^{4}+\cdots \end{aligned}}} and η(τ) is the Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups
Baby_monster_group
exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon
T-group_(mathematics)
Mathematical function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane
Dedekind_eta_function
axiom Dedekind completeness Dedekind cut Dedekind discriminant theorem Dedekind domain Dedekind eta function Dedekind function Dedekind group Dedekind number
List of things named after Richard Dedekind
List_of_things_named_after_Richard_Dedekind
Type of infinite group in group theory
1998). "On Generalized Dedekind Groups and Tarski Super Monsters". Journal of Algebra. 226. A. Yu. Olshanskii, An infinite group with subgroups of prime
Tarski_monster_group
Topics referred to by the same term
that visits each vertex in a graph exactly once Hamiltonian group, a non-abelian Dedekind group in algebra Hamilton Rating Scale for Depression Hamilton
Hamilton
Set whose pairs have minima and maxima
y} have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition. A lattice ( L , ≤ ) {\displaystyle (L,\leq )} is called
Lattice_(order)
In mathematics, Dedekind sums are certain finite sums of products of a sawtooth function. Dedekind introduced them in the 1880's to express the functional
Dedekind_sum
Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens: G is a Dedekind group, or G contains an abelian
Iwasawa_group
Orientation-preserving mapping class group of the torus
credited to Richard Dedekind, in reference to (Dedekind 1877). The map of groups (2, 3, ∞) → (2, 3, n) (from modular group to triangle group) can be visualized
Modular_group
Axioms for the natural numbers
mathematical logic, the Peano axioms (/piˈɑːnoʊ/; [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers
Peano_axioms
Sporadic simple group
\end{aligned}}} and η(τ) is the Dedekind eta function. Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol
Fischer_group_Fi22
Generalization of the Riemann zeta function for algebraic number fields
In mathematics, the Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function
Dedekind_zeta_function
Sporadic simple group
{1}{q}}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes
Fischer_group_Fi23
Non-abelian group of order eight
embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations. Richard Dedekind considered the field Q ( 2
Quaternion_group
Sporadic simple group
+3345q^{3}+12256q^{4}+39350q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Norton & Wilson (1986) found the 14 conjugacy classes of
Harada–Norton_group
Abelian group in which every element can, in some sense, be divided by positive integers
R modules coincide with the divisible R modules if and only if R is a Dedekind domain. Injective object Injective module Pure subgroup Griffith, p.6 Hall
Divisible_group
Sporadic simple group
{2}+681q^{3}+1956q^{4}+5135q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. It can be defined in terms of the generators a and b and
Held_group
Four finite groups derived from the Leech lattice
{1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Griess, p. 97. Thomas Thompson, pp. 148–152. Conway & Sloane
Conway_group
Branch of mathematics that studies algebraic structures
group Dedekind group, Hamiltonian group Examples of groups Trivial group Additive group Permutation group Symmetric group Alternating group p-group List
List of abstract algebra topics
List_of_abstract_algebra_topics
Group of matrices with determinant 1
special linear group over a field or a Euclidean domain is generated by transvections, and the stable special linear group over a Dedekind domain is generated
Special_linear_group
Mathematical concept
In mathematics, a ring is said to be a Dedekind-finite ring (also called directly finite rings and Von Neumann finite rings) if ab = 1 implies ba = 1 for
Dedekind-finite_ring
Function in algebra
ring is RP. The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero
Valuation_(algebra)
Mathematical conjecture
Tate relating the K 2 {\displaystyle K_{2}} group of a field to its Dedekind zeta function. Formally, the group K 2 {\displaystyle K_{2}} of a number field
Birch–Tate_conjecture
Number representing a continuous quantity
ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered
Real_number
Sporadic simple group
\end{aligned}}} and η(τ) is the Dedekind eta function. Conway et al. (1985) "ATLAS: Conway group Co3". "ATLAS: Conway group Co1". "ATLAS: Co3 — Permutation
Conway_group_Co3
Algebra of formal sums
subgroup. Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind was a precursor to the analogous Nielsen–Schreier
Free_abelian_group
Group with series of normal subgroups where all factors are cyclic
the lattice of subgroups of a group, and is sometimes called the Jordan–Dedekind chain condition. Moreover, a finite group is supersolvable if and only
Supersolvable_group
Group with translationally invariant total order
group multiplicatively, this may be shown by considering the Dedekind completion, G ^ {\displaystyle {\widehat {G}}} of the closure of a l.o. group under
Linearly_ordered_group
transformation Transfer (group theory) N. Abel M. Aschbacher R. Baer R. Brauer W. Burnside R. Carter A. Cauchy A. Cayley J.H. Conway R. Dedekind L.E. Dickson M
List_of_group_theory_topics
Mathematical group
called a Bianchi orbifold. An exact formula for its volume in terms of the Dedekind zeta function of the underlying imaginary quadratic field was computed
Bianchi_group
Concept in mathematics
one-dimensional additive group corresponding to each simple root. This statement is false without the pinning; for example, suppose that A is a Dedekind domain and
Reductive_group
Projective variety that is also an algebraic group
field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain
Abelian_variety
cannot carry a group structure is from sets X {\displaystyle X} with the following two properties: X {\displaystyle X} is an infinite Dedekind-finite set
Group structure and the axiom of choice
Group_structure_and_the_axiom_of_choice
Concept in group theory
contains the center of the group. It is contained inside the second term of the upper central series. It is a Dedekind group, so is either abelian or has
Baer_norm
Finite extension of the rationals
a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers. For general Dedekind rings
Algebraic_number_field
Filtration of the Galois group of a local field extension
extension L of K. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when L/K
Ramification_group
Group of mathematical theorems
McLarty, Colin (2006), "Emmy Noether's "Set Theoretic" Topology: From Dedekind to the Rise of Functors", in Gray, Jeremy; Ferreirós, José (eds.), The
Isomorphism_theorems
Submodule of fractions in abstract algebra
context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like
Fractional_ideal
Generalizations of codimension-1 subvarieties of algebraic varieties
back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve
Divisor_(algebraic_geometry)
In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or
Fundamental_group_scheme
German mathematician (1902–1979)
American Mathematical Society 58: 390–419 MR 0015108 Capable group Dedekind group Retract (group theory) Radical of a ring Semiprime ring Nielsen–Schreier
Reinhold_Baer
Subject area in mathematics
K0(A) is isomorphic to Z, by rank. For A a Dedekind domain, K0(A) = Pic(A) ⊕ Z, where Pic(A) is the Picard group of A, An algebro-geometric variant of this
Algebraic_K-theory
Type of lattice in mathematical order theory
groups of module type (Dualgruppen vom Modultypus). He also proved that the modular identity and its dual are equivalent. In the same paper, Dedekind
Modular_lattice
Algebraic structure with addition and multiplication
contributions by Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether. Rings were first formalized as a generalization of Dedekind domains that occur
Ring_(mathematics)
2016 studio album by Dedekind Cut
studio album by American experimental artist Fred Warmsley, under the alias Dedekind Cut. It was released on 11 November 2016, by NON Worldwide and Hospital
Successor_(album)
Mathematical connection between field theory and group theory
ISBN 978-0-486-45868-7. Scharlau, Winfried; Dedekind, Ilse; Dedekind, Richard (1981). Richard Dedekind 1831–1981; eine Würdigung zu seinem 150. Geburtstag
Galois_theory
Second-order theory of the real numbers
is a linearly ordered abelian group under addition with distinguished positive element 1, and that this group is Dedekind-complete, divisible, and Archimedean
Tarski's axiomatization of the reals
Tarski's_axiomatization_of_the_reals
Type of classification in algebra
Archimedean group has the property that, for every Dedekind cut of the group, and every group element ε > 0, there exists another group element x with
Archimedean_group
Branch of mathematics
also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator
Abstract_algebra
Formula in number theory
embeddings of K. ζK(s) is the Dedekind zeta function of K. hK is the class number, the number of elements in the ideal class group of K. RegK is the regulator
Class_number_formula
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
algebraic surfaces. The Picard group of the spectrum of a Dedekind domain is its ideal class group. The invertible sheaves on projective space Pn(k) for k
Picard_group
Branch of number theory
single object, the idele class group, that describes both the quotient by this lattice and the ideal class group. The Dedekind zeta function of a number field
Algebraic_number_theory
Mathematical model
of fractions K of a Dedekind domain R is the "push-forward" of AK from Spec(K) to Spec(R), in other words the "best possible" group scheme AR defined over
Néron_model
In mathematics, an arithmetic surface over a Dedekind domain R {\displaystyle R} with fraction field K {\displaystyle K} is a geometric object having one
Arithmetic_surface
linearly ordered abelian group under addition with distinguished element 1. R {\displaystyle \mathbb {R} } is also Dedekind-complete and divisible. We
Construction of the real numbers
Construction_of_the_real_numbers
Type of mathematical group
{\displaystyle \zeta _{F}} its Dedekind zeta function. Let Γ O {\displaystyle \Gamma _{\mathcal {O}}} be the arithmetic group obtained from O {\displaystyle
Arithmetic_Fuchsian_group
Type of Dirichlet series associated to number field extensions
ideals, that for the trivial representation of the Galois group, the Artin L-function is the Dedekind zeta function of the smaller field: L ( s , 1 , L / K
Artin_L-function
Mathematical conjecture about zeros of L-functions
associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Several textbooks of number theory
The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund
Vorlesungen über Zahlentheorie
Vorlesungen_über_Zahlentheorie
K-theory of fields, and Dedekind zeta-functions" (PDF). Bull. AMS. pp. 155–162. Neumann, W.D. (2004). "Extended Bloch group and the Cheeger-Chern-Simons
Bloch_group
Algebraic structure with addition, multiplication, and division
conceived neither an explicit notion of a field, nor of a group. In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed
Field_(mathematics)
Seventh letter in the Greek alphabet
η-reduction in lambda calculus. Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function. In category theory, the unit
Eta
for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882. If OK is the ring of integers of K, and tr denotes the field trace
Different_ideal
Mathematician (1845–1918)
mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on holiday in Gersau in Switzerland
Georg_Cantor
Conjecture on zeros of the zeta function
hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. Since Dedekind zeta function for abelian extension of the
Riemann_hypothesis
Natural number
number 24 appears prominently in the theory of modular forms through the Dedekind eta function η ( τ ) = q 1 / 24 ∏ n > 0 ( 1 − q n ) , q = e 2 π i τ . {\displaystyle
24_(number)
Algebraic structure
they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains
Principal_ideal_domain
Natural number
168 is a Dedekind number, and an idoneal numbers, and a Cunningham number. 168 is the order of the second smallest nonabelian simple group P S L ( 2
168_(number)
Subset of incomparable elements
Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of
Antichain
Branch of mathematics that studies sets
study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly
Set_theory
Dedekind domain, M is completely decomposable (with respect to a suitable basis) as a direct sum of fractional ideals. Every lattice over a Dedekind domain
Lattice_(module)
Statement in abstract algebra
ideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R. However, over a Dedekind domain the ideal
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
2018 studio album by Dedekind Cut
second studio album by American musician Fred Warmsley, under the alias Dedekind Cut. It was released on February 23, 2018, by Kranky. Tahoe was met with
Tahoe_(album)
Theorem that every subgroup of a free group is itself free
non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian. Jakob Nielsen (1921) originally
Nielsen–Schreier_theorem
Measure of the size of the ring of integers
several important analytic formulas such as the functional equation of the Dedekind zeta function of K {\displaystyle K} , and the analytic class number formula
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Mathematician and philosopher (1906–1978)
foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor. Gödel's discoveries in the foundations of mathematics
Kurt_Gödel
Type of character in number theory
L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous
Hecke_character
Algebraic construction
number field is the unique maximal order in the field. It is always a Dedekind domain. The ring of integers OK is a finitely-generated Z {\displaystyle
Ring_of_integers
Alternative decimal expansion of 1
0.999... = 1. The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. The above approach to assigning a real
0.999...
Index of lists with the same name
double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function of a number field Duursma zeta function of error-correcting
List_of_zeta_functions
Field of knowledge
the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure
Mathematics
Theorem relating a group with the image and kernel of a homomorphism
back to the work of Richard Dedekind, and was further formalized by Emmy Noether into the isomorphism theorems. Given two groups G {\displaystyle G} and H
Fundamental theorem on homomorphisms
Fundamental_theorem_on_homomorphisms
Number system extending the rational numbers
fields, in an analogous way. This will be described now. Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of
P-adic_number
Concept in abstract algebra
valuation ring with a value group isomorphic to the integers under addition. R {\displaystyle R} is a local ring, a Dedekind domain, and not a field. R
Discrete_valuation_ring
Elements in exactly one of two sets
abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being
Symmetric_difference
Meromorphic function on the complex plane
Most notably, the mathematicians Bernhard Riemann (1826–1866), Richard Dedekind (1831–1916), Erich Hecke (1887–1947) and Emil Artin (1898–1962) investigated
L-function
the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta function. Casas-Alvero conjecture:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Italian mathematician (1853–1925)
real numbers, an area in which he extended the research begun by Richard Dedekind. Completing his high school studies privately at only 16 years of age,
Gregorio_Ricci-Curbastro
Submodule of a mathematical ring
There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The related, but distinct
Ideal_(ring_theory)
Mathematical concept
works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets. Dedekind's approach was essentially to adopt the idea
Infinity
Number used for counting
Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in
Natural_number
fashions by gentle circles. The Grobianus Tischzucht (1538) and Friedrich Dedekind's Grobianus (1549) works give their name to Grobianism. The poet representatives
List of poetry groups and movements
List_of_poetry_groups_and_movements
Algebraic object with an ordered structure
an ordered subfield that is isomorphic to the rational numbers. Every Dedekind-complete ordered field is isomorphic to the reals. Squares are necessarily
Ordered_field
Arithmetic operation
set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a Dedekind cut of rationals: a non-empty
Addition
Number n where n and totient(n) are coprime
An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. Any prime number is clearly cyclic. All cyclic numbers
Cyclic_number_(group_theory)
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DEDEKIND GROUP
DEDEKIND GROUP
DEDEKIND GROUP
DEDEKIND GROUP
DEDEKIND GROUP
DEDEKIND GROUP
DEDEKIND GROUP
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DEDEKIND GROUP
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