Cyclic group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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 $G=⟨g⟩$.

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.

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 $n\mapsto g^n$

Arithmetic functions

See finite cyclic group#Arithmetic functions and group of integers#Arithmetic functions.

Particular cases

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$n$	Cyclic group of order $n$
$1$	Trivial group
$2$	Cyclic group:Z2
$3$	Cyclic group:Z3
$4$	Cyclic group:Z4
$5$	Cyclic group:Z5
$6$	Cyclic group:Z6
$7$	Cyclic group:Z7
$8$	Cyclic group:Z8
$9$	Cyclic group:Z9

Relation with other 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

Stronger properties

• Finite cyclic group
• Group of prime order
• Odd-order cyclic group

Weaker properties

• Abelian group
• Metacyclic group
• Polycyclic group
• Locally cyclic group
• Aut-abelian group: For proof of the implication, refer cyclic implies aut-abelian and for proof of its strictness (i.e. the reverse implication being false) refer aut-abelian not implies cyclic.
• Group of nilpotency class two
• Nilpotent group
• Supersolvable group
• Solvable group

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 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. 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 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. 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

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 54 (formal definition)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 3 (definition introduced in paragraph)
• Topics in Algebra by I. N. Herstein, ^{More info}, Example 2.4.3, Page 39 (definition introduced in example)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 9
• An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 47
• Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 2
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 9
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 33 (defined as cyclic subgroup)
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 46, leading to point (2.7) (defined as cyclic subgroup) and Page 47, Point (2.9)

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