Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently <math>\Z</math>n or Zn, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertible 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 of the group.

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 integers Z, with the operation of addition, forms a group.

This is the multiplicative group of units of the ring Z/nZ; there are φ(n) of them, where again φ is the Euler totient function. 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/nZ)× is cyclic, its generators are called primitive roots modulo n.

For a prime number p, the group (Z/pZ)× is always cyclic, consisting of the non-zero elements of the finite field of order p. More generally, every finite subgroup of the multiplicative group of any field is cyclic.

Rotational symmetries

The set of rotational symmetries of a polygon 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/nZ. 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 rotoreflections.

The group of all rotations of a circle (the circle group, also denoted S1) is not cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C∞ is countable, while S1 is not. The group of rotations by rational angles is countable, but still not cyclic.

Galois theory

An nth root of unity is a complex number whose nth power is 1, a root of the polynomial . The set of all nth roots of unity forms a cyclic group of order n under multiplication. 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 subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form ⟨m⟩ = mZ, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they 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. Thus, since a prime number p has no nontrivial divisors, pZ is a maximal proper subgroup, and the quotient group Z/pZ is simple; in fact, a cyclic group is simple if and only if its order is prime.

All quotient groups Z/nZ are finite, with the exception For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, generated by the residue class of n/d. There are no other subgroups.

Additional properties

Every cyclic group is abelian. (And observe that when n is prime, there is exactly one element whose order is a proper divisor of n, namely the identity.)

The order of an element m in Z/nZ is n/gcd(n,m).

If n and m are coprime, then the direct product of two cyclic groups Z/nZ and Z/mZ is isomorphic to the cyclic group Z/nmZ, 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, then any group with p elements is isomorphic to the simple group Z/pZ.

A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when . The sequence of cyclic numbers include all primes, but some are composite 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 and for finite n.

Associated objects

Representations

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.

Cycle graph

A cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, 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 but is usually suppressed. Z2 is sometimes drawn with two curved edges as a multigraph.

A cyclic group Zn, with order n, corresponds to a single cycle graphed simply as an n-sided polygon with the elements at the vertices.

{| class="wikitable"

|+ Cycle graphs up to order 24

|- align=center

| 40px

| 60px

|- align=center

| Z1|| Z2|| Z3|| Z4|| Z5|| Z6 = Z3×Z2|| Z7|| Z8

|- align=center

| 60px

|- align=center

| Z9||Z10 = Z5×Z2||Z11||Z12 = Z4×Z3||Z13||Z14 = Z7×Z2||Z15 = Z5×Z3||Z16

|- align=center

| 60px

|- align=center

| Z17||Z18 = Z9×Z2||Z19||Z20 = Z5×Z4||Z21 = Z7×Z3||Z22 = Z11×Z2||Z23||Z24 = Z8×Z3

Cayley graph

thumb|240px|The [[Paley graph of order 13, a circulant graph formed as the Cayley graph of Z/13 with generator set {1,3,4}]]

A Cayley graph 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, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. 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 graphs whose symmetry group includes a transitive cyclic group.

Endomorphisms

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)×. an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n. Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.

Procyclic groups

A profinite group is called procyclic if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers <math>\widehat{\Z}</math> or the p-adic integers <math>\Z_p</math> for a prime number p.

Locally cyclic groups

A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, 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 is a distributive lattice.

Cyclically ordered groups

A cyclically ordered group is a group together with a cyclic order 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 is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. 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 or nilpotent group is polycyclic.

Examples

Integer and modular addition

Rotational symmetries

Galois theory

Subgroups

Additional properties

Associated objects

Representations

Cycle graph

Cayley graph

Endomorphisms

Procyclic groups

Locally cyclic groups

Cyclically ordered groups

Metacyclic and polycyclic groups

See also

Footnotes

Notes

Citations

References

Further reading

External links