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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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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

Contents

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

Particular cases

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)

External links

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