In
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 groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
in pure
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ...
of
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
, 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 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 language ...
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 commut ...
(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 o ...
), and every
finitely generated abelian group is a
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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 it ...
, 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
![Cyclic group](https://upload.wikimedia.org/wikipedia/commons/5/5f/Cyclic_group.svg)
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 ...
forms a group under multiplication. It is cyclic, since it is generated by the
primitive root 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 C
6 = ⟨''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 C
3 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, 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, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
.
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 symmetri ...
. 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 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 language ...
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 ot ...
.)
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 ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
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 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 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
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 ot ...
. 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, 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 ...
of order ''p''. More generally, every finite
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 ...
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 toge ...
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 const ...
(the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
, 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 Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
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 form ...
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 sur ...
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 exa ...
. 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 pol ...
of the
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 ration ...
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 solvabl ...
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 aro ...
, 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
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 ...
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 ''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 ...
s and
quotient group
A quotient group or factor group is a mathematical 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 "factored" out). For examp ...
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, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 union, a ...
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
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 o ...
: (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 wor ...
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
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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 mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
. 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 fixed ...
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 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 ...
and is particularly useful in visualizing the structure of small
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s. 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. Z
2 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
![Paley13](https://upload.wikimedia.org/wikipedia/commons/0/05/Paley13.svg)
A
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 Cayle ...
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 terminal ...
. 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 i ...
s whose
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
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 p ...
of the abelian group Z/''n''Z is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
(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
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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 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
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 ration ...
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, et ...
, 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
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 union, a ...
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 uni ...
.
Cyclically ordered groups
A
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. In ...
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
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
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
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 product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
s 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\;\overs ...
. 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, 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 commut ...
or
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 .
Intuiti ...
is polycyclic.
See also
*
Cycle graph (group)
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cyclic group, cycles of a group (mathematics), group and is particularly useful in visualizing the structure of small finite groups.
A cycle is the Set ...
*
Cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-mod ...
*
Cyclic sieving In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.
Definition
Let ''C'' be a cyclic group gene ...
*
Prüfer group
In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots.
...
(
countably infinite
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; ...
analogue)
*
Circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
(
uncountably infinite
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
analogue)
Footnotes
Notes
Citations
References
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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 GroupNamesEvery cyclic group is abelian
{{DEFAULTSORT:Cyclic Group
Abelian group theory
Properties of groups