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Group theory
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In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive). Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

Contents

Definition

A group G is called cyclic if there exists an element g in G such that G = <g> = { gn | n is an integer }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains g is G itself suffices to show that G is cyclic. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers For example, 1 + 2 = 3 (mod 6) corresponds to g1·g2 = g3, and 2 + 5 = 1 (mod 6) corresponds to g2·g5 = g7 = g1, and so on. One can use the isomorphism φ defined by φ(gi) = i.

For every positive integer n there is exactly one cyclic group (up to isomorphism) whose order is n, and there is exactly one infinite cyclic group (the integers under addition). Hence, the cyclic groups are the simplest groups and they are completely classified.

The name 'cyclic' may be misleading: it is possible to generate infinitely many elements and not form any literal cycles; that is, every gn is distinct. (It can be said that it has one infinitely long cycle. ) A group generated in this way is called an infinite cyclic group, and is isomorphic to the additive group of integers Z. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

Since the cyclic groups are abelian, they are often written additively and denoted Zn. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the However, this notation can be problematic for number theorists because it conflicts with the usual notation for p-adic number rings or localization at a prime ideal. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G We adopt the first of these here to avoid the collision of notation. See also the section Subgroups and notation below.

One may write the group multiplicatively, and denote it by Cn, where n is the order (which can be ∞). For example, g3g4 = g2 in C5, whereas 3 + 4 = 2 in Z/5Z.

Properties

The fundamental theorem of cyclic groups states that if G is a cyclic group of order n then every subgroup of G is cyclic. In Abstract algebra, the fundamental theorem of cyclic groups states that if G\ is a Cyclic group of order n\ then every Subgroup In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of Moreover, the order of any subgroup of G is a divisor of n and for each positive divisor k of n the group G has exactly one subgroup of order k. This property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor d of n the group has at most one subgroup of order d. Sometimes the equivalent statement is used: a group of order n is cyclic if and only if for every divisor d of n the group has exactly one subgroup of order d.

Every finite cyclic group is isomorphic to the group { [0], [1], [2], . In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective . . , [n − 1] } of integers modulo n under addition, and any infinite cyclic group is isomorphic to Z (the set of all integers) under addition. Thus, one only needs to look at such groups to understand the properties of cyclic groups in general. Hence, cyclic groups are one of the simplest groups to study and a number of nice properties are known.

Given a cyclic group G of order n (n may be infinity) and for every g in G,

More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without The order of the residue class of m is n / gcd(n,m). In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group Cp or Z/pZ. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in

The direct product of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than Thus e. g. Z/12Z is the direct product of Z/3Z and Z/4Z, but not the direct product of Z/6Z and Z/2Z.

The definition immediately implies that cyclic groups have very simple group presentation C∞ = < x | > and Cn = < x | xn > for finite n. In Mathematics, one method of defining a group is by a presentation.

A primary cyclic group is a group of the form Z/pk where p is a prime number. A primary cyclic group is a Cyclic group of prime power order C_{p^m}\ (for any prime p, and Natural number In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1

Z/nZ and Z are also commutative rings. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property If p is a prime, then Z/pZ is a finite field, also denoted by Fp or GF(p). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Every field with p elements is isomorphic to this one. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

The units of the ring Z/nZ are the numbers coprime to n. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than They form a group under multiplication modulo n with φ(n) elements (see above). In Modular arithmetic the set of Congruence classes Relatively prime to the modulus n form a group under multiplication called the multiplicative It is written as (Z/nZ)×. For example, we get (Z/nZ)× = {1,5} when n = 6, and get (Z/nZ)× = {1,3,5,7} when n = 8.

In fact, it is known that (Z/nZ)× is cyclic if and only if n is 2 or 4 or pk or 2 pk for an odd prime number p and k ≥ 1, in which case every generator of (Z/nZ)× is called a primitive root modulo n. In Mathematics, the parity of an object states whether it is even or odd In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Modular arithmetic, a branch of Number theory, a primitive root modulo n is any number g with the property that any number Coprime Thus, (Z/nZ)× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2

The group (Z/pZ)× is cyclic with p − 1 elements for every prime p, and is also written (Z/pZ)* because it consists of the non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Examples

In 2D and 3D the symmetry group for n-fold rotational symmetry is Cn, of abstract group type Zn. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. In 3D there are also other symmetry groups which are algebraically the same, see Cyclic symmetry groups in 3D. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

Note that the group S1 of all rotations of a circle (the circle group) is not cyclic, since it is not even countable. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex

The nth roots of unity form a cyclic group of order n under multiplication. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power e. g. , 0 = z3 − 1 = (z − s0)(z − s1)(z − s2) where si = e2πi / 3 and a group of {s0,s1,s2} under multiplication is cyclic.

The Galois group of every finite field extension of a finite field is finite and cyclic; 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. In Mathematics, a Galois group is a group associated with a certain type of Field extension. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

Representation

The cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices. In Group theory, a sub-field of Abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful The dark vertex in the cycle graphs below stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.

C1 C2 C3 C4 C5 C6 C7 C8

The representation theory of the cyclic group is a critical base case for the representation theory of more general finite groups. In Mathematics, Representation theory is a technique for analyzing abstract groups in terms of groups of Linear transformations See the article on 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. This article refers to the use of the term character theory in mathematics for the media studies definition see Character theory (Media. 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 subgroups and more generally the representation theory of blocks of cyclic defect. Modular representation theory is a branch of Mathematics, and is that part of Representation theory which studies Linear representations of Finite group In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which

Subgroups and notation

All subgroups and quotient groups of cyclic groups are cyclic. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Specifically, all subgroups of Z are of the form mZ, with m an integer ≥0. All of these subgroups are different, and apart from the trivial group (for m=0) all are isomorphic to Z. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. In Mathematics, the lattice of subgroups of a group G is the lattice whose elements are the Subgroups of G with the In the mathematical area of Order theory, every Partially ordered set P gives rise to a dual (or opposite) partially ordered set which In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without All factor groups of Z are finite, except for the trivial exception Z/{0} = Z/0Z. For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, the one generated by the residue class of n/d. There are no other subgroups. The lattice of subgroups is thus isomorphic to the set of divisors of n, ordered by divisibility. In particular, a cyclic group is simple if and only if its order (the number of its elements) is prime. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning

Using the quotient group formalism, Z/nZ is a standard notation for the additive cyclic group with n elements. In ring terminology, the subgroup nZ is also the ideal (n), so the quotient can also be written Z/(n) or Z/n without abuse of notation. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. These alternatives does not conflict with the notation for the p-adic integers. The last form has the additional advantage that it reads the same way that the group or ring is often described verbally, "Zee mod en".

As a practical problem, one may be given a finite subgroup C of order n, generated by an element g, and asked to find the size m of the subgroup generated by gk for some integer k. Here m will be the smallest integer > 0 such that mk is divisible by n. It is therefore n/m where m = (k, n) is the gcd of k and n. Put another way, the index of the subgroup generated by gk is m. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH This reasoning is known as the index calculus algorithm, in number theory. In Group theory, the index calculus algorithm is an Algorithm for computing Discrete logarithms This is the best known algorithm for certain groups such Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes

Endomorphisms

The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Under this isomorphism, the number r corresponds to the endomorphism of Z/nZ 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 of Z/nZ is isomorphic to the unit group (Z/nZ)× (see above). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, i. e. to {−1, +1} \cong C2.

Virtually cyclic groups

A group is called virtually cyclic if it contains a cyclic subgroup of finite index. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH It is known that a finitely generated discrete group with exactly two ends is virtually cyclic. In Mathematics, a discrete group is a group G equipped with the Discrete topology. Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic. In Group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated

See also

References

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