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: 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.

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

No.	Shorthand	A group is termed cyclic (sometimes, monogenic or monogenous) if ...	A group $G$ is termed cyclic if ...
1	modular arithmetic definition	it is either isomorphic to the group of integers or to the group of integers modulo n for some positive integer $n$. Note that the case $n = 1$ gives the trivial group.	$G \cong \mathbb{Z}$ or $G \cong \mathbb{Z}/n\mathbb{Z}$ for some positive integer $n$. Note that the case $n = 1$ gives the trivial group.
2	generating set of size one	it has a generating set of size 1.	there exists a $g \in G$ such that $G = \langle g \rangle$.
3	quotient of group of integers	it is isomorphic to a quotient of the group of integers	it is isomorphic to a quotient group of the group of integers $\mathbb{Z}$, i.e., there exists a surjective homomorphism from $\mathbb{Z}$ to $G$.

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	cyclicity is subgroup-closed	If $G$ is a cyclic group and $H$ is a subgroup of $G$, $H$ is also a cyclic group.
quotient-closed group property	Yes	cyclicity is quotient-closed	If $G$ is a cyclic group and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is also a cyclic group.
finite direct product-closed group property	No	cyclicity is not finite direct product-closed	It is possible to have cyclic groups $G_1$ and $G_2$ such that the external direct product $G_1 \times G_2$ is not a cyclic group. In fact, if both $G_1$ and $G_2$ are nontrivial finite cyclic groups and their orders are not relatively prime to each other, or if one of them is infinite, the direct product will not be cyclic.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite cyclic group	both cyclic and a finite group			|FULL LIST, MORE INFO
group of prime order				Finite cyclic group|FULL LIST, MORE INFO
odd-order cyclic group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group	any two elements commute	cyclic implies abelian	abelian not implies cyclic	Epabelian group, Locally cyclic group, Residually cyclic group|FULL LIST, MORE INFO
metacyclic group	has a cyclic normal subgroup with a cyclic quotient group	(obvious)	metacyclic not implies cyclic	Characteristically metacyclic group, Group with metacyclic derived series|FULL LIST, MORE INFO
polycyclic group	has a subnormal series where all the successive quotient groups are cyclic groups			Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic group|FULL LIST, MORE INFO
locally cyclic group	every finitely generated subgroup is cyclic			|FULL LIST, MORE INFO
group whose automorphism group is abelian		cyclic implies abelian automorphism group	abelian automorphism group not implies abelian	Locally cyclic group|FULL LIST, MORE INFO
group of nilpotency class two	the inner automorphism group is abelian	(via abelian)	(via abelian)	Group whose automorphism group is abelian, Group whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
nilpotent group		(via abelian)	(via abelian)	Abelian group, Epinilpotent group, Group of nilpotency class two, Group whose automorphism group is nilpotent|FULL LIST, MORE INFO
finitely generated group	has a finite generating set	cyclic means abelian with a generating set of size one	any finite non-cyclic group such as the Klein four-group	Polycyclic group|FULL LIST, MORE INFO
finitely generated abelian group	finitely generated and abelian	follows from separate implications for finitely generated and abelian	Klein four-group is a counterexample.	|FULL LIST, MORE INFO
finitely generated nilpotent group	finitely generated and nilpotent	(via finitely generated abelian)	(via finitely generated abelian)	Finitely generated abelian group|FULL LIST, MORE INFO
supersolvable group		(via finitely generated abelian)	(via finitely generated abelian)	Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic group|FULL LIST, MORE INFO
solvable group				Abelian group, Metabelian group, Metacyclic group, Nilpotent group, Polycyclic group|FULL LIST, MORE INFO

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

References

Textbook references

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

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Definition links

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