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
Definition
No. | Shorthand | A group is termed cyclic (sometimes, monogenic or monogenous) if ... | A group 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 . Note that the case gives the trivial group. | or for some positive integer . Note that the case gives the trivial group. |
2 | generating set of size one | it has a generating set of size 1. | there exists a such that . |
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 , i.e., there exists a surjective homomorphism from to . |
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 under a surjective homomorphism from to must generate
- Conversely, if an element generates , we get a surjective homomorphism by
Arithmetic functions
See finite cyclic group#Arithmetic functions and group of integers#Arithmetic functions.
Particular cases
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Cyclic group of order | |
---|---|
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 is a cyclic group and is a subgroup of , is also a cyclic group. |
quotient-closed group property | Yes | cyclicity is quotient-closed | If is a cyclic group and is a normal subgroup of , the quotient group 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 and such that the external direct product is not a cyclic group. In fact, if both and 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
Facts
- There is exactly one cyclic group (upto isomorphism of groups) of every positive integer order : namely, the group of integers modulo . There is a unique infinite cyclic group, namely
- 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 978-0471433347^{More info} | 54 | formal definition | ||
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} | 3 | definition introduced in paragraph | ||
Topics in Algebra by I. N. Herstein^{More info} | 39 | Example 2.4.3 | definition introduced in example | |
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} | 9 | |||
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444^{More info} | 47 | |||
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info} | 2 | |||
Algebra by Serge Lang, ISBN 038795385X^{More info} | 9 | |||
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} | 33 | defined as cyclic subgroup | ||
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More info} | 46 | Page 46: leading to point (2.7), Page 47, Point (2.9) |
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