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Algebraic structure
the binary cyclic group of the n-gon is the cyclic group of order 2n, C 2 n {\displaystyle C_{2n}} , thought of as an extension of the cyclic group C n
Binary_cyclic_group
Nonabelian group of order 120
3-sphere by the binary icosahedral group. binary polyhedral group binary cyclic group, ⟨n⟩, order 2n binary dihedral group, ⟨2,2,n⟩, order 4n binary tetrahedral
Binary_icosahedral_group
Nonabelian group in algebraic group theory
that cyclically rotates i, j, and k. One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all
Binary_tetrahedral_group
polyhedral group binary cyclic group, ⟨n⟩, index 2n binary dihedral group, ⟨2,2,n⟩, index 4n binary tetrahedral group, 2T=⟨2,3,3⟩, index 24 binary icosahedral
Binary_octahedral_group
Type of cyclic group in group theory
extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension
Dicyclic_group
Mathematical group that can be generated as the set of powers of a single element
In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused
Cyclic_group
Groups of point isometries in 3 dimensions
n {\displaystyle A_{n}} : binary cyclic group of an (n + 1)-gon, order 2n D n {\displaystyle D_{n}} : binary dihedral group of an n-gon, ⟨2,2,n⟩, order
Point groups in three dimensions
Point_groups_in_three_dimensions
the group within that order. Common group names: Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of
List_of_small_groups
Error-detecting code for detecting data changes
information) and the algorithm is based on cyclic codes. CRCs are popular because they are simple to implement in binary hardware, easy to analyze mathematically
Cyclic_redundancy_check
Ordering of binary values, used for positioning and error correction
code". Other names include "cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB). The Gray code is
Gray_code
Mathematical abelian group
four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group
Klein_four-group
Concept in geometry
U1(H): the binary cyclic groups, binary dihedral groups, binary tetrahedral group, binary octahedral group, and binary icosahedral group. Hoggar, S.G
Quaternionic_polytope
Alternative mathematical ordering
cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation
Cyclic_order
Integer whose multiples are digit rotations
cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely-known cyclic number
Cyclic_number
Type of linear error-correcting code
out ∞.) As a cyclic code: The perfect G23 code can be constructed via the factorization of x 23 + 1 {\displaystyle x^{23}+1} over the binary field GF(2):
Binary_Golay_code
Open conjecture that no real circulant Hadamard matrix has order greater than 4
discrete Fourier transforms, Littlewood polynomials, perfect binary sequences, and cyclic Menon–Hadamard difference sets. It is also closely connected
Ryser's conjecture on circulant Hadamard matrices
Ryser's_conjecture_on_circulant_Hadamard_matrices
Mathematical structure with multiplication as its operation
multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication
Multiplicative_group
System of digitally encoding numbers
(xxiv+835+1 pages) (NB. Ledley classified the described cyclic code as a cyclic decimal-coded binary code.) Savard, John J. G. (2018) [2006]. "Decimal Representations"
Binary-coded_decimal
there are essentially only 18 possibilities. One of these is C3, the cyclic group with three elements. The others all have a semigroup with two elements
Semigroup_with_three_elements
Subclass of manifold
3, and 3. The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group (of order 48) which has the
Spherical_3-manifold
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
Non-abelian group of order eight
subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. Another characterization is that a finite p-group in which there
Quaternion_group
a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message
Computation of cyclic redundancy checks
Computation_of_cyclic_redundancy_checks
16-element matrix group
central product of a cyclic group of order 4 and the dihedral group of order 8. The Pauli group is a representation of the gamma group in three-dimensional
Pauli_group
Symmetry of molecules of chemical compounds
divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one
Molecular_symmetry
Classification system for crystals
notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: Cn (for cyclic) indicates that
Crystallographic_point_group
Group obtained by aggregating similar elements of a larger group
the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of
Quotient_group
subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic. Every locally cyclic group is abelian
Glossary_of_group_theory
Mathematical concept
groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups
Direct_product_of_groups
Seemingly random, difficult to predict bit stream created by a deterministic algorithm
A pseudorandom binary sequence (PRBS), pseudorandom binary code or pseudorandom bitstream is a binary sequence that, while generated with a deterministic
Pseudorandom_binary_sequence
Type of classification in algebra
is that an Archimedean group is a linearly ordered group without any bounded cyclic subgroups: there does not exist a cyclic subgroup S and an element
Archimedean_group
Number n where n and totient(n) are coprime
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 are square-free.
Cyclic_number_(group_theory)
Set with associative invertible operation
the three axioms. Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the
Group_(mathematics)
Index of articles associated with the same name
Order in mathematics may refer to: Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary
Order_(mathematics)
Sporadic simple group
to expand the group PΓL(3,4) to M24. The group M24 also is the permutation automorphism group of the binary Golay code W, i.e., the group of permutations
Mathieu_group_M24
Chemical compound with hydrogen and carbon group atoms
used in the production of PVC. The other group 14 elements have a lower tendency to catenate. Hydrosilicons (binary silicon-hydrogen compounds), a silicon
Group_14_hydride
Double cover Lie group of the special orthogonal group
these latter, given a cyclic group of odd order Z 2 k + 1 {\displaystyle \mathrm {Z} _{2k+1}} in SO(n), its preimage is a cyclic group of twice the order
Spin_group
Sequence of digital values used for synchronisation
inventor Ronald Hugh Barker. The process is described in "Group Synchronisation of Binary Digital Systems" published in 1953. These sequences were initially
Barker_code
133-dimensional exceptional simple Lie group
fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is
E7_(mathematics)
Example of a Semigroup
the semigroup (Z2, +2) is given below: This group is isomorphic to the cyclic group Z2 and the symmetric group S2. Let A be the three-element set {1, 2,
Semigroup_with_two_elements
Collection of groups
classes of groups are: ∅ {\displaystyle \emptyset } : the empty class of groups C {\displaystyle {\mathfrak {C}}~} : the class of cyclic groups A {\displaystyle
Class_of_groups
Group of unitary complex matrices with determinant of 1
algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } , and is composed
Special_unitary_group
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral
Covering groups of the alternating and symmetric groups
Covering_groups_of_the_alternating_and_symmetric_groups
Exponentation in modular arithmetic
D) mod c Diffie–Hellman key exchange uses exponentiation in finite cyclic groups. The above methods for modular matrix exponentiation clearly extend
Modular_exponentiation
Relationship between elements of two sets
In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the
Binary_relation
Generalised alphabetical order
the lexicographical order. For instance, the set of countably infinite binary sequences (by definition, the set of functions from natural numbers to {
Lexicographic_order
Mathematical term in group theory
now usually presented as a group of automorphisms of the infinite regular binary rooted tree. The study of Grigorchuk's group informed in large part the
Grigorchuk_group
Natural number, cyclic number
preceding 142,858. It is both a Kaprekar number and a cyclic number. 142857 is the best-known cyclic number in base 10, being the six repeating digits of
142857
Algebraic structure with an associative operation and an identity element
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the natural numbers with addition
Monoid
Quadratic homogeneous polynomial in two variables
_{2}(\mathbb {Z} )} . When f is definite, the group is finite, and when f is indefinite, it is infinite and cyclic. A binary quadratic form q ( x , y ) {\displaystyle
Binary_quadratic_form
constructions of such codes include using cyclic reversible extended generalized Hadamard matrices, and a binary approach. Before diving into these constructions
Coding theory approaches to nucleic acid design
Coding_theory_approaches_to_nucleic_acid_design
Class of astronomical objects
A symbiotic binary is a type of binary star system, often simply called a symbiotic star. They usually contain a white dwarf with a companion red giant
Symbiotic_binary
Binary representation for signed numbers
negative, and zero) integers on computers, and more generally, fixed point binary values. As with the ones' complement and sign-magnitude systems, two's complement
Two's_complement
Subset of a group that forms a group itself
in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More
Subgroup
Conversion process for computer data
so-called "binary-object storage framework", which support serialization into and retrieval from a compact binary form. Both handle cyclic, recursive
Serialization
Algebraic structure
algebraic structure consisting of a set together with an associative internal binary operation on it. In mathematical analysis, the term also appears in the
Semigroup
unique group with exactly 4 elements, as there are in this case two non-isomorphic groups in total: the cyclic group of order 4 and the Klein four-group. There
Essentially_unique
In group theory, equivalence class under the relation of conjugation
fact, isomorphic either to a cyclic group of order p 2 {\displaystyle p^{2}} or to the direct product of two cyclic groups of order p . {\displaystyle
Conjugacy_class
Operation on the subsets of a set
a single element, that is, the closure of this element, is called a cyclic group. In linear algebra, the closure of a non-empty subset of a vector space
Closure_(mathematics)
6-carbon simple sugar
molecular weight is 180.156 g/mol. Hexoses exist in two forms, open-chain or cyclic, that easily convert into each other in aqueous solutions. The open-chain
Hexose
In mathematics a group is a set together with a binary operation on the set (usually called multiplication) that obeys the group axioms. The axiom of
Group structure and the axiom of choice
Group_structure_and_the_axiom_of_choice
Sporadic simple group
certain relation, certain products of commuting involutions generate the binary Golay code, which extends to the maximal subgroup 211:M24. Bolt, Bray, and
Janko_group_J4
Four finite groups derived from the Leech lattice
its automorphism group Co0. Conway started his investigation of Co0 with a subgroup he called N, a holomorph of the (extended) binary Golay code (as diagonal
Conway_group
Base-16 numeric representation
hardware is binary in nature and that hex is power of 2, the hex representation is often used in computing as a dense representation of binary information
Hexadecimal
In number theory, measure of non-unique factorization
does not possess unique factorization; in fact the class group of R {\displaystyle R} is cyclic of order 2. Indeed, the ideal J = ( 2 , 1 + − 5 ) {\displaystyle
Ideal_class_group
algorithm Belief propagation Berger code Berlekamp–Massey algorithm Binary Golay code Binary Goppa code Bipolar violation CRHF Casting out nines Check digit
List of algebraic coding theory topics
List_of_algebraic_coding_theory_topics
A quadratic residue code is a type of cyclic code. Examples of quadratic residue codes include the ( 7 , 4 ) {\displaystyle (7,4)} Hamming code over G
Quadratic_residue_code
Classification of pitch class sets
correspond to the binary bracelets of length 12: that is, equivalence classes of binary sequences of length 12 under the operations of cyclic permutation and
Forte_number
Complementary binary-coded decimal code
Aiken code (also known as 2421 code) is a complementary binary-coded decimal (BCD) code. A group of four bits is assigned to the decimal digits from 0 to
Aiken_code
Well-quasi-ordering of finite trees
t e Order theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order
Kruskal's_tree_theorem
Chemical compound with hydrogen and pnictogen atoms
Pnictogen hydrides or hydrogen pnictides are binary compounds of hydrogen with pnictogen atoms (elements of group 15: nitrogen, phosphorus, arsenic, antimony
Pnictogen_hydride
Five sporadic simple groups
by Conway and Sloane. The group M24 is the permutation automorphism group of the extended binary Golay code W, i.e., the group of permutations on the 24
Mathieu_group
Order-preserving mathematical function
coefficient - measure of monotonicity in a set of data Total monotonicity Cyclical monotonicity Operator monotone function Monotone set function Absolutely
Monotonic_function
Mathematical operation
of binary relations, the composition of relations is the forming of a new binary relation R ; S {\displaystyle R\mathbin {;} S} from two given binary relations
Composition_of_relations
Group that is a topological space with continuous group operations
same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse
Topological_group
3D symmetry group
icosahedral group (subject of this article, also known as H3) 2I, the binary icosahedral group They correspond to the following short exact sequences (the latter
Icosahedral_symmetry
Mathematical tool in group theory
Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal
Cayley_table
Device storing number of times an event or process occurred
state. The state indicates the current count, encoded directly as a binary or binary-coded decimal (BCD) number or using encodings such as one-hot or Gray
Counter_(digital)
Mathematical puzzle game
all disks at peg B is reached halfway, i.e. after (3n − 1) / 2 moves. In Cyclic Hanoi, we are given three pegs (A, B, C), which are arranged as a circle
Tower_of_Hanoi
Number that permute or shift cyclically when multiplied by another number
or shift cyclically when they are multiplied by another integer n {\displaystyle n} . Examples are: 142857 × 3 = 428571 (shifts cyclically one place
Transposable_integer
Recursive integer sequence
binary operator can be represented in terms of a full binary tree, by labeling each leaf a, b, c, d. It follows that Cn is the number of full binary trees
Catalan_number
Chemical compounds containing only hydrogen and one other chemical element
many group 14 hydrides. Going down the group the number of binary silicon compounds (silanes) is small (straight or branched but rarely cyclic) for example
Binary_compounds_of_hydrogen
Any binary chemical compound containing just silicon and another chemical element
containing silicon bonded to a more electropositive element. Binary silicon compounds can be grouped into several classes. Saltlike silicides are formed with
Binary_compounds_of_silicon
Algebraic structure
{\displaystyle C_{n}\ltimes C_{q^{n}-1}} , where C k {\displaystyle C_{k}} is the cyclic group of order k {\displaystyle k} . The divisibility conditions on n {\displaystyle
Near-field_(mathematics)
Law (all real numbers are positive, negative, or 0)
every real number is either positive, negative, or zero. More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one
Law_of_trichotomy
Topic in group theory
{\displaystyle S_{2}\wr S_{2}\wr \cdots \wr S_{2}} is the automorphism group of a complete binary tree. Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur
Wreath_product
Use of mathematical groups in magnetochemistry
to the cyclic groups, the binary dihedral groups, the binary tetrahedral group, the binary octahedral group, and the binary icosahedral group; and for
Finite_subgroups_of_SU(2)
Binary relation over a set and itself
binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary)
Homogeneous_relation
Decision tracking and managing method
elements, using a symbol (or the figure '1'). Such marking is defined as Binary DSM. The marking then has developed to indicate quantitative relation Numeric
Design_structure_matrix
On chains and antichains in partial orders
t e Order theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order
Dilworth's_theorem
Class of error-correcting code
linear codes include: Repetition code Parity code Cyclic code Hamming code Golay code, both the binary and ternary versions Polynomial codes, of which BCH
Linear_code
Data used to detect errors in other data
algorithms most used in practice, such as Fletcher's checksum, Adler-32, and cyclic redundancy checks (CRCs), address these weaknesses by considering not only
Checksum
group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic)
Fundamental unit (number theory)
Fundamental_unit_(number_theory)
Theorem in order theory
t e Order theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order
Dushnik–Miller_theorem
Plane tiling corresponding to a polyhedron
symmetry group of the spherical polyhedron is the product of its rotation group (the symmetry group of the projective polyhedron) and the cyclic group of order
Projective_polyhedron
Partially ordered vector space, ordered as a lattice
order is a non-Archimedean Riesz space. Riesz spaces are lattice ordered groups Every Riesz space is a distributive lattice Convex cone – Mathematical set
Riesz_space
Minimal polynomial of a primitive element in a finite field
divides x4 − 1: its roots generate a cyclic group of order 4, while the multiplicative group of GF(32) is a cyclic group of order 8. The polynomial x2 + 2x
Primitive polynomial (field theory)
Primitive_polynomial_(field_theory)
Group of symmetries of a regular polygon
to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group. The following Cayley table
Dihedral_group
Star in the constellation Canis Minor
notable for its proximity to the prominent star Procyon. This is a probable binary star system. β Canis Minoris, which is Latinised to Beta Canis Minoris,
Beta_Canis_Minoris
There are equally many countable order types and real numbers
t e Order theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order
Cantor–Bernstein_theorem
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
Boy/Male
Indian, Punjabi, Sikh
Blessing
Girl/Female
Indian
Modesty
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Girl/Female
Hindu
Shore, Musical instrument, Goddess of wealth
Boy/Male
Anglo, British, English
With Royal Might
Surname or Lastname
English (chiefly South Yorkshire)
English (chiefly South Yorkshire) : topographic name for someone who lived on land enclosed by a bend in a river, from Old English binnan ēa ‘within the river’, or a habitational name from places in Kent called Binney and Binny, which have this origin.Scottish : habitational name from Binney or Binniehill near Falkirk, named in Gaelic as Beinnach, from beinn ‘hill’ + the locative suffix -ach.
Female
Turkish
Turkish name PINAR means "spring."
Female
English
English pet form of German Belinda, possibly BINDY means "bright serpent" or "bright linden tree."
Female
Hebrew
(×‘Ö¼Ö´×™× Ö¸×”) Hebrew name BINA means "intelligence, wisdom."Â
Male
Hindi/Indian
Variant spelling of Hindi Vijay, BIJAY means "victory."
Male
Hindi/Indian
(विनय) Hindi name VINAY means "leading asunder."
Boy/Male
English
royal.
Girl/Female
English
Originally a diminutive used for names ending in -bina, like Albina, Columbina, and Robina, now...
Female
Hebrew
Variant spelling of Hebrew Bina, BINAH means "intelligence, wisdom."Â
Male
English
English unisex form of Latin Hilarius and Hilaria, HILARY means "joyful; happy."Â Originally, this was strictly a masculine name.
Boy/Male
American, Australian, French, German, Greek, Latin, Polish, Swedish
Cheerful; Happy; Joyful; Similar to Hilary
Girl/Female
Indian
(the wife of Sage Kashyap)
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Boy/Male
Indian
An intimate particle of the God of heaven
Surname or Lastname
English
English : variant spelling of Vickery.
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
Girl/Female
Hindu
River Yamuna, Surya putri Yamuna
Boy/Male
Sikh
Enjoys cleanliness
Girl/Female
Danish, Indian
Cute; Beautiful
Male
Hebrew
(×—Ö²× Ö·× Ö°×™Ö¸×”ï¬µ) Variant form of Hebrew Chananya, CHANANYAHU means "whom Jehovah has graciously given."
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Lord Vishnu; Lord Shiva
Boy/Male
Indian
Boy/Male
Indian
Voice from Heart; Gift of Heart
Male
French
Variant spelling of French Alphonse, ALFONSE means "noble and ready."
Boy/Male
Muslim
Goodness
Girl/Female
American, British, English, Greek
Pure; Keeper of the Keys
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
BINARY CYCLIC-GROUP
a.
Of or pertaining to the urine; as, the urinary bladder; urinary excretions.
n.
Wine made in the Canary Islands; sack.
v. i.
To pass through a cycle of changes; to recur in cycles.
a.
Relating or belonging to bile; conveying bile; as, biliary acids; biliary ducts.
n.
See Finery.
n.
A pale yellow color, like that of a canary bird.
n.
The act or practice of using a cycle; cycling.
n.
A canary bird.
n.
A cycler.
a.
Containing ten; tenfold; proceeding by tens; as, the denary, or decimal, scale.
imp. & p. p.
of Cycle
a.
Of or pertaining to a cycle or circle; moving in cycles; as, cyclical time.
a.
Of a pale yellowish color; as, Canary stone.
a.
Adhering to a fixed circle of legends; cyclic; hence, mean; inferior. See Cyclic poets, under Cyclic.
n.
A binary compound of silicon, or one regarded as binary.
a.
Of or pertaining to the Canary Islands; as, canary wine; canary birds.
p. pr. & vb. n.
of Cycle
a.
Containing cysts; cystose; as, cystic sarcoma.
a.
Alt. of Cyclical
v. i.
To perform the canary dance; to move nimbly; to caper.