Cyclic group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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

Symbol-free definition

A group is said to be cyclic if it satisfies the following equivalent conditions:

• It occurs as a quotient group of the group of integers
• It has a generating set of size 1
• There is an element in the group such that every element in the group can be expressed as a power of that element

Definition with symbols

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Relation with other properties

Weaker properties

• Abelian group
• Metacyclic group
• Polycyclic group

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

• Category: Variations of cyclicity gives group properties obtained by slight variation/modifications to the property of being cyclic

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of 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.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of 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.

Direct product-closedness

A direct product of cyclic groups need not be cyclic. It is cyclic if adn only if the two groups have relatively prime orders.