group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, a branch of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

in pure mathematics, a cyclic group or monogenous group is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

, denoted C_''n'', that is generated by a single element. That is, it is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

elements with a single associative binary operation, 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 an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''

generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...

'' of the group. Every infinite cyclic group is 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 structures ...

of Z, the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

(meaning that its group operation is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

), and every finitely generated abelian group is a direct product 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 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 because the only ways ...

order is a

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

, 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 result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...

, 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 the

subgroup In group theory, a branch of mathematics, 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 precisely, ''H'' is a subgroup ...

that consists of all its integer powers: , called the cyclic subgroup generated by ''g''. The order of ''g'' is the number of elements in ⟨''g''⟩; that is, the order of an element is equal to the order of the cyclic subgroup that it generates. A ''cyclic group'' is a group which is equal to one of its cyclic subgroups: for some element ''g'', called a ''generator'' of ''G''. For a finite cyclic group ''G'' of order ''n'' we have , where ''e'' is the identity element and whenever (

mod Mod, MOD or mods may refer to: Places * Modesto City–County Airport, Stanislaus County, California, US Arts, entertainment, and media Music * Mods (band), a Norwegian rock band * M.O.D. (Method of Destruction), a band from New York City, US ...

''n''); in particular , and . An abstract group defined by this multiplication is often denoted C_''n'', and we say that ''G'' is 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 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, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...

G = \left\

forms a group under multiplication. It is cyclic, since it is generated by the primitive root

z = \tfrac 1 2 + \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 ''g''^''j'' · ''g''^''k'' = ''g''^''j''+''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 ''z''^''k'' and ''g''^''k'' 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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

notations Z/''n''Z, Z/(''n''), or Z/''n'', some authors denote a finite cyclic group as Z_''n'', but this clashes with the notation of

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, 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. On the other hand, in an infinite cyclic group , the powers ''g''^''k'' give distinct elements for all integers ''k'', so that ''G'' = , and ''G'' is isomorphic to the standard group and to Z, the additive group of the integers. An example is the first

frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...

. 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

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 mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

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 In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In o ...

.) Every finite cyclic group ''G'' is isomorphic to Z/''n''Z, where ''n'' = 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 rings, also denoted Z and Z/''n''Z or Z/(''n''). If ''p'' is a

, then Z/''pZ'' is a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

, 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, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...

of the ring Z/''n''Z; there are ''φ''(''n'') of them, where again ''φ'' is the

. 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

of order ''p''. More generally, every finite

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 grass ...

is cyclic.

Rotational symmetries

The set of

rotational symmetries Rotational symmetry, also known as radial symmetry in geometry, 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 i ...

of a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

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, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...

s. The group of all rotations of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

(the circle group, 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, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

, 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 In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...

is a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

whose ''n''th power is 1, a

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...

of the

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

. The set of all ''n''th roots of unity forms 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, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

of the field extension of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

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 extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...

if its Galois group is cyclic. For fields of

characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...

, such extensions are the subject of

Kummer theory In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer ar ...

, and are intimately related to

solvability by radicals In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

. For an extension of

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

s and quotient groups 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 to Z. The

lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their uni ...

of Z is isomorphic to the 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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...

of ''n''/''d''. There are no other subgroups.

Additional properties

Every cyclic group is abelian. That is, its group operation is

: (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, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...

''p''^''k'', the group Z/''p''^''k''Z is called a

primary cyclic group In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number ''p''. That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and natural number ''m''. Every f ...

. 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, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...

es consists of a single element. A cyclic group of order ''n'' therefore has ''n'' conjugacy classes. If ''d'' is a

divisor 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 ...

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''/ gcd(''n'',''m''). If ''n'' and ''m'' are

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

, then the direct product 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. For example, Z/12Z is isomorphic to the direct product under the isomorphism ; but it is not isomorphic to , 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 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 because the only ways ...

, then any group with ''p'' elements is isomorphic to the simple group Z/''p''Z. A number ''n'' is called a

cyclic number A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 :14 ...

if Z/''n''Z is the only group of order ''n'', which is true exactly when . The sequence of 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. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

and for finite ''n''.

Associated objects

Representations

The

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

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 In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

s and more generally the representation theory of blocks of cyclic defect.

Cycle graph

A cycle graph illustrates the various cycles of a

and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a

circular graph In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...

, 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. A trivial path (identity) can be drawn as a

loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, an ...

but is usually suppressed. Z₂ is sometimes drawn with two curved edges as a

multigraph In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more ...

. A cyclic group 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

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...

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 that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...

, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite

path graph In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two termina ...

. 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 graphs. 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 mathematical field of graph theory, a vertex-transitive graph is a graph in which, given any two vertices and of , there is some automorphism :f : G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive ...

s whose symmetry group includes a transitive cyclic group.

Endomorphisms

The

endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...

of the abelian group Z/''n''Z is isomorphic to Z/''n''Z itself as a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

.. 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 consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

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 index (the number of

coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

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 End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games ** End (gridiron footbal ...

; an example of such a group is the direct product 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

locally cyclic group In mathematics, a locally cyclic group is a group (''G'', *) in which every finitely generated subgroup is cyclic. Some facts * Every cyclic group is locally cyclic, and every locally cyclic group is abelian. * Every finitely-generated locally c ...

is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the

s: every finite set of rational numbers is a set of integer multiples of a single

unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...

, the inverse of their

lowest common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The low ...

, 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

is a

distributive lattice In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...

Cyclically ordered groups

cyclically ordered group In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger ...

is a group together with a

cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...

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

metacyclic group In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ''G'' for which there is a short exact sequence :1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\, where ''H'' and ''K'' ar ...

is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include the cyclic groups, the

dicyclic group In group theory, a dicyclic group (notation Dic''n'' or Q4''n'', Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST) is a particular kind of non-abelian group of order 4''n'' (''n'' > 1). It is an extension of the ...

s, and the direct products of two cyclic groups.The

polycyclic group In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational ...

s generalize metacyclic groups by allowing more than one level of

group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...

. 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

nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intui ...

is polycyclic.

Footnotes

Notes

Citations

References

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

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

Every cyclic group is abelian
{{DEFAULTSORT:Cyclic Group Abelian group theory Properties of groups