Cyclic group

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).

Definition

That is, we say G is cyclic if there exists an element g in G such that G = <g> = { gⁿ for all integers n }. 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 = { e, g¹, g², g³, g⁴, g⁵ }, then G is cyclic, and, G is essentially the same as (that is, isomorphic to) the group of { 0, 1, 2, 3, 4, 5 } with addition modulo 6. I.e. 1 + 2 mod 6 = 3, 2 + 5 mod 6 = 1, and so on. One can use the isomorphism φ defined by φ(g) = 1.

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.

Unlike the name suggests, it is possible to generate infinitely many elements and not form any literal cycles; that is, every $g^{n}$ is distinct. A group generated in this way is called an infinite cyclic group, every one of which is isomorphic to the additive group of integers Z.

Since the groups are abelian they are often written additively, and denoted by Z_n; however, this notation is often avoided by number theorists because it conflicts or is easily confused with the usual notation for p-adic number rings or localisation at a prime ideal. The quotient notation Z/n or Z/nZ (see also below) is an alternative.

One may write the group multiplicatively, and denote it by C_n. (For example, g³g⁴ = g² in C₅, whereas 3 + 4 = 2 (mod 5) in Z/5Z.)

All finite cyclic groups are periodic groups.

Properties

Every cyclic group is isomorphic to the group { 0, 1, 2, ..., n − 1 } under addition modulo n, or Z, the additive group of all of integers. Thus, one only needs to look at such groups to understand 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,

• G is abelian; that is, their group operation is commutative: gh = hg. This is so since g + h mod n = h + g mod n.
• If n is finite, then $g^{n}=e$ since n mod n = 0.
• If n = ∞, then there are exactly two generators: namely 1 and −1 for Z, and any others mapped to them under an isomorphism in other infinite cyclic groups.
• If n is finite, then there are exactly φ(n) generators where φ() is the Euler phi function
• Every subgroup of G is cyclic. 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.
• C_n is isomorphic to Z/nZ (factor group of Z over nZ) since Z/nZ = {0 + nZ, 1 + nZ, 2 + nZ, 3 + nZ, 4 + nZ, ..., n − 1 + nZ} $\cong$ { 0, 1, 2, 3, 4, ..., n − 1} under addition modulo n.

The generators of Z/nZ are the residue classes of the integers which are coprime to n; the number of those generators is known as φ(n), where φ is Euler's totient function.

More generally, if d is a divisor of n, then the number of elements in Z/nZ 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 Z_p.

The direct product of two cyclic groups Z_n and Z_m is cyclic if and only if n and m are coprime. Thus e.g. Z₁₂ is the direct product of Z₃ and Z₄, but not of Z₆ and Z₂.

Immediately from the definition we know that cyclic groups have very simple presentation of the form < x | xⁿ >

The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many cyclic groups.

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

The units of the ring Z_n are the numbers coprime to n. They form a group under multiplication modulo n; it has φ(n) elements (see above). It is written as Z_n^×.

For example, we get Z_n^× = {1,5} when n = 6, and get Z_n^× = {1,3,5,7} when n = 8.

In fact, it is known that Z_n^× is cyclic if and only if n is 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_n^× is called a primitive root modulo n.

Thus, Z_n^× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group.

The group Z_p^× is cyclic with p − 1 elements for every prime p. More generally, every finite subgroup of the multiplicative group of any field is cyclic.

Examples

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 cyclic symmetry groups in 3D.

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., $0=z^{3}-1=(z-s^{0})(z-s^{1})(z-s^{2})$ where $s^{i}=e^{2\pi i/3}$ and a group of $\{s^{0},s^{1},s^{2}\}$ 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

The cycle graphs of finite cyclic groups are all n-sided 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.

Z1	Z2	Z3	Z4	Z5	Z6	Z7	Z8

Subgroups and notation

All subgroups and factor 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}. For every positive divisor d of n, the 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 ring terminology, the subgroup nZ is also the ideal (n), so the quotient can also be written Z/(n) or even Z/n without abuse of notation. The last form has the advantages that it reads exactly the same way that the group or ring is often described verbally, "Zee mod en"; and it does not conflict with the notation for p-adic integers.

In particular: a cyclic group is simple if and only if its order (the number of its elements) is prime.

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 g^k 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 g^k is m. This reasoning is known as the index calculus algorithm, in number theory.

Endomorphisms

The endomorphism ring of the abelian group Z_n is isomorphic to Z_n itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z_n which 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_n is isomorphic to the group Z_n^× (see above). The automorphism group of Z_n is sometimes called the character group of Z_n and the construction of this group leads directly to the definition of Dirichlet characters.

Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z, and its automorphism group is isomorphic to the group of units of the ring Z, i.e. to {−1, +1} $\cong$ Z₂.

See also

• Cyclic symmetry groups in 3D
• Cyclic extension
• Cyclic module
• Modular arithmetic