Cyclic group

From Groupprops

Jump to: navigation, search

Contents

This article is about a basic definition in group theory. The article text may, however, contain more material. Rate its utility as a basic definition article on the talk page
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in group theory | Go through a guided tour for beginners to this wiki
This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties
VIEW RELATED: | | Group metaproperty satisfactions | | |
RANDOM GROUP PROPERTY: Simple group: A nontrivial group having no proper nontrivial normal subgroup.

This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

Definition

Definition in terms of modular arithmetic

A group is said to be cyclic (sometimes, monogenic or monogenous) if it is either isomorphic to the group of integers or to the group of integers modulo n for some positive integer n.

In symbols, G is a cyclic group if and only if G \cong \mathbb{Z} or G \cong \mathbb{Z}/n\mathbb{Z} for some positive integer n. Note that the case n = 1 gives the trivial group.

Since the group of integers mod n has order n, a cyclic group isomorphic to this group is termed the cyclic group of order n.

Definition in terms of generating sets

A group is termed cyclic (sometimes, monogenic or monogenous) if it has a generating set of size 1.

In symbols, a group G is termed cyclic if there exists a g \in G such that G = \langle g \rangle.

Such an element g is termed a cyclic element or generator for G.

Definition as a quotient

A group is termed cyclic if it is a quotient of the group \mathbb{Z}, in other words, there exists a surjective homomorphism from \mathbb{Z} to the group.

A group G is termed cyclic (sometimes, monogenic or monogenous) if it satisfies the following equivalent conditions:

Equivalence of definitions

Further information: Equivalence of definitions of cyclic group

The second and third definition are equivalent because the subgroup generated by an element is precisely the set of its powers. The first definition is equivalent to the other two, because:

  • The image of 1 \in \mathbb{Z} under a surjective homomorphism from \mathbb{Z} to G must generate G
  • Conversely, if an element g generates G, we get a surjective homomorphism \mathbb{Z} \to G by n \mapsto g^n

Relation with other properties

Weaker properties

Related properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Facts

  • There is exactly one cyclic group (upto isomorphism of groups) of every positive integer order n: namely, the group of integers modulo n. There is a unique infinite cyclic group, namely \mathbb{Z}
  • For any group and any element in it, we can consider the subgroup generated by that element. That subgroup is, by definition, a cyclic group. Thus, every group is a union of cyclic subgroups. Further information: Every group is a union of cyclic subgroups

Metaproperties

Subgroups

This group property is subgroup-closed, viz any subgroup of a group satisfying the property also satisfies the property
View other subgroup-closed group properties

Any subgroup of a cyclic group is cyclic. In fact, given a finite cyclic group, there is a unique subgroup of each order which is cyclic. For full proof, refer: Cyclicity is subgroup-closed

Quotients

This group property is quotient-closed, viz any quotient of a group satisfying the property also has the property
View other quotient-closed group properties

Any quotient of a cyclic group is cyclic. The generator for this is the image of the generator for the original group, under the quotient map. For full proof, refer: Cyclicity is quotient-closed

Direct product-closedness

A direct product of cyclic groups need not be cyclic. It is cyclic if and only if the two groups have relatively prime orders. For full proof, refer: Cyclicity is not direct product-closed

References

Textbook references

External links

Search for "cyclic+group" on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo, Windows Live
Learn more about using the Searchbox OR provide your feedback

Definition links

Facts about Cyclic groupRDF feed
Defined inDummitFoote (?, ?, ?)  +, AlperinBell (?, ?, ?)  +, Herstein (?, ?, ?)  +, RobinsonGT (?, ?, ?)  +, RobinsonAA (?, ?, ?)  +, FGTAsch (?, ?, ?)  +, Lang (?, ?, ?)  +, Hungerford (?, ?, ?)  +, Artin (?, ?, ?)  +, Wikipedia (?, ?, ?)  +, Planetmath (?, ?, ?)  +, and Mathworld (?, ?, ?)  +
Referenced inDummitFoote (?, ?, ?)  +, AlperinBell (?, ?, ?)  +, Herstein (?, ?, ?)  +, RobinsonGT (?, ?, ?)  +, RobinsonAA (?, ?, ?)  +, FGTAsch (?, ?, ?)  +, Lang (?, ?, ?)  +, Hungerford (?, ?, ?)  +, Artin (?, ?, ?)  +, Wikipedia (?, ?, ?)  +, Planetmath (?, ?, ?)  +, and Mathworld (?, ?, ?)  +
Stronger thanAbelian group  +, Metacyclic group  +, and Polycyclic group  +
Retrieved from "http://groupprops.subwiki.org/wiki/Cyclic_group"
Personal tools