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In algebra, a cyclic group is a group that is generated by a single element, in the sense that every element of the group can be written as a power of some particular element g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a "generator" of the group. Any infinite cyclic group is isomorphic to Z, the integers with addition as the group operation. Any finite cyclic group of order n is isomorphic to Z/nZ, the integers modulo n with addition as the group operation.

1 Definition
2 Properties
3 Examples
4 Representation
5 Subgroups and notation
6 Endomorphisms
7 Virtually cyclic groups
8 See also
9 Notes
10 References
11 External links
12 Further reading

Definition [edit]

The six 6th complex roots of unity form a cyclic group under multiplication. z is a primitive element, but z² is not, because the odd powers of z are not a power of z².

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.

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. 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 χ defined by χ(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.

Furthermore, the circle group (whose elements are uncountable) is not a cyclic group—a cyclic group always has countable elements.

Since the cyclic groups are abelian, they are often written additively and denoted Z_n. 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. The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. 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 C_n, where n is the order (which can be ∞). For example, g²g⁴ = g¹ in C₅, whereas 2 + 4 = 1 in Z/5Z.

Properties [edit]

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

G is abelian; that is, their group operation is commutative: gh = hg (for all g and h in G). This is so since r + s ≡ s + r (mod n).
If n is finite, then gⁿ = g⁰ is the identity element of the group, since kn ≡ 0 (mod n) for any integer k.
If n = ∞, then there are exactly two elements that each generate the group: namely 1 and −1 for Z.
If n is finite, then it is isomorphic to the group { [0], [1], [2], ..., [n − 1] } of integers modulo n under addition and there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient function.
Every subgroup of G is cyclic. (see fundamental theorem of cyclic groups and see also a section below) Indeed, each finite subgroup of G is a group of { 0, 1, 2, 3, ..., m − 1 } with addition modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to (so isomorphic to) Z.
Every quotient group of G is cyclic. In fact, under any group homomorphism, the image of a cyclic group is generated by the image of a generator of the cyclic group.

More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). The order of the residue class of m is n / gcd(n,m).

If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group C_p or Z/pZ. There are more numbers with the same property, see cyclic number.

The direct product of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime. 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 group presentation C_∞ = ⟨x |⟩ and C_n = ⟨x | xⁿ⟩ for finite n.

A primary cyclic group is a group of the form Z/p^kZ where p is a prime number. 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.

Z/nZ and Z are also commutative rings. If p is a prime, then Z/pZ is a finite field, also denoted by F_p or GF(p). Every field with p elements is isomorphic to this one.

The units of the ring Z/nZ are the numbers coprime to n. They form a group under multiplication modulo n with φ(n) elements (see above). It is written as (Z/nZ)^×. For example, when n = 6, we get (Z/nZ)^× = {1,5}. When n = 8, we get (Z/nZ)^× = {1,3,5,7}.

In fact, it is known that (Z/nZ)^× is cyclic if and only if n is 1 or 2 or 4 or p^k or 2 p^k for an odd prime number p and k ≥ 1, in which case every generator of (Z/nZ)^× is called a primitive root modulo n. Thus, (Z/nZ)^× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group.

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. For example, this follows from the characterization below.

Let G be a finite group. Then G is a cyclic group if, for each n > 0, G contains at most n elements of order dividing n.^{[citation needed]}

Examples [edit]

In 2D and 3D the symmetry group for n-fold rotational symmetry is C_n, of abstract group type Z_n. In 3D there are also other symmetry groups which are algebraically the same, see Symmetry groups in 3D that are cyclic as abstract group.

Note that the group S¹ of all rotations of a circle (the circle group) is not cyclic, since it is not even countable.

The n^th roots of unity form a cyclic group of order n under multiplication. e.g., 0 = z³ − 1 = (z − s⁰)(z − s¹)(z − s²) where s = e^2πi/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.

Representation [edit]

The cycle graphs of finite cyclic groups are all n-sided regular polygons with the elements at the vertices. 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 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 subgroups and more generally the representation theory of blocks of cyclic defect.

Subgroups and notation [edit]

All subgroups and quotient groups of cyclic groups are cyclic. 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. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. 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.^[1]

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. These alternatives do not conflict with the notation for the p-adic integers. The last form is very common in informal calculations; it has the additional advantage that it reads the same way that the group or ring is often described verbally in English, "Zee mod en".

Endomorphisms [edit]

The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. 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).

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 [edit]

A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by applying a member of the cyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. It is known that a finitely generated discrete group with exactly two ends is virtually cyclic (for instance the product of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.

Notes [edit]

^ Gannon (2006), p. 18

References [edit]

Gallian, Joseph (1998), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin, ISBN 978-0-669-86179-2 , especially chapter 4.
Herstein, I. N. (1996), Abstract algebra (3rd ed.), Prentice Hall, ISBN 978-0-13-374562-7, MR 1375019 , especially pages 53–60.
Gannon, Terry (2006), Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics, Cambridge monographs on mathematical physics, Cambridge University Press, ISBN 978-0-521-83531-2
Milne, Group theory, http://www.jmilne.org/math/CourseNotes/gt.html

External links [edit]

Hazewinkel, Michiel, ed. (2001), "Cyclic group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
An introduction to cyclic groups

Cyclic group

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