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In
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, a branch of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a cyclic group or monogenous group is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
that is generated by a single element. That is, it is a set of invertible elements with a single
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as a power of ''g'' in multiplicative notation, or as a multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of Z, the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. Every finite cyclic group of
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
(meaning that its group operation is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
), and every finitely generated abelian group is a
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of cyclic groups. Every cyclic group of
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
order is a
simple group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, which cannot be broken down into smaller groups. In the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.


Definition and notation

For any element ''g'' in any group ''G'', one can form a
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 precisely ...
of all integer powers ⟨''g''⟩ = , called a cyclic subgroup of ''g''. The
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
of ''g'' is the number of elements in ⟨''g''⟩; that is, the order of an element is equal to the order of its cyclic subgroup. A ''cyclic group'' is a group which is equal to one of its cyclic subgroups: for some element ''g'', called a ''generator''. For a finite cyclic group ''G'' of order ''n'' we have ''G'' = , where ''e'' is the identity element and ''g''''i'' = ''g''''j'' whenever ''i'' ≡ ''j'' ( mod ''n''); in particular ''g''''n'' = ''g''0 = ''e'', and ''g''−1 = ''g''''n''−1. An abstract group defined by this multiplication is often denoted C''n,'' and we say that ''G'' is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to the standard cyclic group C''n''. Such a group is also isomorphic to Z/''n''Z, the group of integers modulo ''n'' with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism ''χ'' defined by ''χ''(''g''''i'') = ''i'' the identity element ''e'' corresponds to 0, products correspond to sums, and powers correspond to multiples. For example, the set of complex 6th
roots of unity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
G = \left\
forms a group under multiplication. It is cyclic, since it is generated by the primitive root z=\tfrac12 +\tfraci=e^: that is, ''G'' = ⟨''z''⟩ = with ''z''6 = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C6 = ⟨''g''⟩ = with multiplication ''gj'' · ''gk'' = ''gj+k'' (mod 6), so that ''g''6 = ''g''0 = e. These groups are also isomorphic to Z/6Z = with the operation of addition modulo 6, with ''zk'' and ''gk'' corresponding to ''k''. For example, corresponds to , and corresponds to , and so on. Any element generates its own cyclic subgroup, such as ⟨''z''2⟩ = of order 3, isomorphic to C3 and Z/3Z; and ⟨''z''5⟩ = = ''G'', so that ''z''5 has order 6 and is an alternative generator of ''G''. Instead of the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
notations Z/''n''Z, Z/(''n''), or Z/''n'', some authors denote a finite cyclic group as Z''n'', but this conflicts with the notation of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, where Z''p'' denotes a ''p''-adic number ring, or
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
at a
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
. On the other hand, in an infinite cyclic group ''G ='' ⟨''g''⟩'','' the powers ''g''''k'' give distinct elements for all integers ''k'', so that ''G'' = , and ''G'' is isomorphic to the standard group C = C and to Z, the additive group of the integers. An example is the first
frieze groupIn mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction. Such patterns occur frequently in architecture and decorative art. A frieze group is the set of symmetry, symmetries of a frieze ...
. Here there are no finite cycles, and the name "cyclic" may be misleading. To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".


Examples


Integer and modular addition

The set of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s Z, with the operation of addition, forms a group. It is an infinite cyclic group, because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to Z. For every positive integer ''n'', the set of integers modulo ''n'', again with the operation of addition, forms a finite cyclic group, denoted Z/''n''Z. A modular integer ''i'' is a generator of this group if ''i'' is
relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
to ''n'', because these elements can generate all other elements of the group through integer addition. (The number of such generators is ''φ''(''n''), where ''φ'' is the
Euler totient function The first thousand values of . The points on the top line represent when is a prime number, which is In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to th ...
.) Every finite cyclic group ''G'' is isomorphic to Z/''n''Z, where ''n'' = , ''G'', is the order of the group. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
s, also denoted Z and Z/''n''Z or Z/(''n''). If ''p'' is a
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then Z/''pZ'' is a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and is usually denoted F''p'' or GF(''p'') for Galois field.


Modular multiplication

For every positive integer ''n'', the set of the integers modulo ''n'' that are relatively prime to ''n'' is written as (Z/''n''Z)×; it forms a group under the operation of multiplication. This group is not always cyclic, but is so whenever ''n'' is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime . This is the multiplicative group of
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
of the ring Z/''n''Z; there are ''φ''(''n'') of them, where again ''φ'' is the
Euler totient function The first thousand values of . The points on the top line represent when is a prime number, which is In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to th ...
. For example, (Z/6Z)× = , and since 6 is twice an odd prime this is a cyclic group. In contrast, (Z/8Z)× = is a Klein 4-group and is not cyclic. When (Z/''n''Z)× is cyclic, its generators are called primitive roots modulo ''n''. For a prime number ''p'', the group (Z/''p''Z)× is always cyclic, consisting of the non-zero elements of the
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of order ''p''. More generally, every finite
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 precisely ...
of the multiplicative group of any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
is cyclic.


Rotational symmetries

The set of
rotational symmetries Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...
of a
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
forms a finite cyclic group. If there are ''n'' different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to Z/''n''Z. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead
rotoreflection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
s. The group of all rotations of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
''S''1 (the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, also denoted ''S''1) is ''not'' cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C is
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, while ''S''1 is not. The group of rotations by rational angles ''is'' countable, but still not cyclic.


Galois theory

An ''n''th
root of unity The 5th roots of unity (blue points) in the complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are containe ...
is a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
whose ''n''th power is 1, a
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
of the
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
''x''''n'' − 1. The set of all ''n''th roots of unity form a cyclic group of order ''n'' under multiplication. For example, the polynomial factors as , where ; the set = forms a cyclic group under multiplication. The
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s generated by the ''n''th roots of unity forms a different group, isomorphic to the multiplicative group (Z/''n''Z)× of order ''φ''(''n''), which is cyclic for some but not all ''n'' (see above). A field extension is called a
cyclic extensionIn abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois ...
if its Galois group is cyclic. For fields of
characteristic zero In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, such extensions are the subject of
Kummer theoryIn abstract algebra and number theory, Kummer theory provides a description of certain types of field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemati ...
, and are intimately related to solvability by radicals. For an extension of
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of characteristic ''p'', its Galois group is always finite and cyclic, generated by a power of the Frobenius mapping. Conversely, given a finite field ''F'' and a finite cyclic group ''G'', there is a finite field extension of ''F'' whose Galois group is ''G''.


Subgroups

All
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 precisely ...
s and
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
s of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form ⟨''m''⟩ = ''m''Z, with ''m'' a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group = 0Z, they all are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to Z. The
lattice of subgroups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of Z is isomorphic to the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of the lattice of natural numbers ordered by
divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
. Thus, since a prime number ''p'' has no nontrivial divisors, ''p''Z is a maximal proper subgroup, and the quotient group Z/''p''Z is
simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
; in fact, a cyclic group is simple if and only if its order is prime. All quotient groups Z/''n''Z are finite, with the exception For every positive divisor ''d'' of ''n'', the quotient group Z/''n''Z has precisely one subgroup of order ''d'', generated by the residue class of ''n''/''d''. There are no other subgroups.


Additional properties

Every cyclic group is abelian. That is, its group operation is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: (for all ''g'' and ''h'' in ''G''). This is clear for the groups of integer and modular addition since , and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order ''n'', ''g''''n'' is the identity element for any element ''g''. This again follows by using the isomorphism to modular addition, since for every integer ''k''. (This is also true for a general group of order ''n'', due to Lagrange's theorem.) For a
prime power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
''pk,'' the group Z/''pk''Z is called a primary cyclic group. The fundamental theorem of abelian groups states that every
finitely generated abelian group In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_ ...
is a finite direct product of primary cyclic and infinite cyclic groups. Because a cyclic group is abelian, each of its
conjugacy class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
es consists of a single element. A cyclic group of order ''n'' therefore has ''n'' conjugacy classes. If ''d'' is a
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

divisor
of ''n'', then the number of elements in Z/''n''Z which have order ''d'' is ''φ''(''d''), and the number of elements whose order divides ''d'' is exactly ''d''. If ''G'' is a finite group in which, for each , ''G'' contains at most ''n'' elements of order dividing ''n'', then ''G'' must be cyclic. The order of an element ''m'' in Z/''n''Z is ''n''/(''n'',''m''). If ''n'' and ''m'' are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, then the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of two cyclic groups Z/''n''Z and Z/''m''Z is isomorphic to the cyclic group Z/''nm''Z, and the converse also holds: this is one form of the
Chinese remainder theorem In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
. For example, Z/12Z is isomorphic to the direct product Z/3Z × Z/4Z under the isomorphism (''k'' mod 12) → (''k'' mod 3, ''k'' mod 4); but it is not isomorphic to Z/6Z × Z/2Z, in which every element has order at most 6. If ''p'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then any group with ''p'' elements is isomorphic to the simple group Z/''p''Z. A number ''n'' is called a cyclic number if Z/''n''Z is the only group of order ''n'', which is true exactly when . The cyclic numbers include all primes, but some are
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials * ...
such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are: :1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... The definition immediately implies that cyclic groups have
group presentation In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...
and for finite ''n''.


Associated objects


Representations

The
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of the cyclic group is a critical base case for the representation theory of more general finite groups. In the complex case, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent. In the positive characteristic case, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic
Sylow subgroup Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norway, Norwegian mathematician who proved foundational results in group theory. Biography He was born and died in Oslo, Christiania (now Oslo). Sylow was a son of governm ...
s and more generally the representation theory of blocks of cyclic defect.


Cycle graph

A cycle graph illustrates the various cycles of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
and is particularly useful in visualizing the structure of small
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. Trivial paths (identity) can be drawn as a loop but are usually suppressed. Z2 is sometimes drawn with two curved edges as a
multigraph In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

multigraph
. A cyclic groups Z''n'', with order ''n'', corresponds to a single cycle graphed simply as an ''n''-sided polygon with the elements at the vertices.


Cayley graph

A
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

Cayley graph
is a graph defined from a pair (''G'',''S'') where ''G'' is a group and ''S'' is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a
cycle graph In graph theory, a cycle graph or circular graph is a Graph (discrete mathematics), graph that consists of a single Cycle (graph theory), cycle, or in other words, some number of vertices (at least 3, if the graph is Simple graph, simple) connect ...

cycle graph
, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite
path graph In the Mathematics, mathematical field of graph theory, a path graph or linear graph is a graph whose vertex (graph theory), vertices can be listed in the order ''v''1, ''v''2, …, ''v'n'' such that the edge (graph theory), edges are where ...
. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called
circulant graph In graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( an ...
s. These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the
vertex-transitive graph In the mathematics, mathematical field of graph theory, a vertex-transitive graph is a Graph (discrete mathematics), graph ''G'' in which, given any two vertices ''v''1 and ''v''2 of ''G'', there is some Graph automorphism, automorphism :f\colon G ...
s whose
symmetry group In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their chang ...
includes a transitive cyclic group.


Endomorphisms

The
endomorphism ringIn abstract algebra, the endomorphisms of an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
of the abelian group Z/''n''Z is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to Z/''n''Z itself as a ring.. Under this isomorphism, the number ''r'' corresponds to the endomorphism of Z/''n''Z that maps each element to the sum of ''r'' copies of it. This is a bijection if and only if ''r'' is coprime with ''n'', so the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
of Z/''n''Z is isomorphic to the unit group (Z/''n''Z)×. Similarly, the endomorphism ring of the additive group of Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, which is .


Related classes of groups

Several other classes of groups have been defined by their relation to the cyclic groups:


Virtually cyclic groups

A group is called virtually cyclic if it contains a cyclic subgroup of finite
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(the number of
coset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two
ends ''Ends'' is a collection of science fiction Science fiction (sometimes shortened to sci-fi or SF) is a genre of speculative fiction which typically deals with imagination, imaginative and futuristic concepts such as advanced science and te ...
; an example of such a group is the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of Z/''n''Z and Z, in which the factor Z has finite index ''n''. Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.


Locally cyclic groups

A
locally cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ern ...
is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its
lattice of subgroups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is a distributive lattice.


Cyclically ordered groups

A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group). Every finite subgroup of a cyclically ordered group is cyclic.


Metacyclic and polycyclic groups

A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a cyclic quotient, ending in the trivial group. Every finitely generated
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
or nilpotent group is polycyclic.


See also

*Cycle graph (group) *Cyclic module *Cyclic sieving *Prüfer group (Countable set, countably infinite analogue) *Circle group (Uncountable set, uncountably infinite analogue)


Footnotes


Notes


Citations


References

* * * * * * * * * * * * * * * * * * *


Further reading

*


External links

*Milne, Group theory, http://www.jmilne.org/math/CourseNotes/gt.html
An introduction to cyclic groups
*
Cyclic groups of small order on GroupNames
{{DEFAULTSORT:Cyclic Group Abelian group theory Properties of groups