# Cyclic group

A cyclic group is a group in which every element is a power of a single element, called a generator: if \(a\) generates \(G\), then every element of \(G\) has the form \(a^i\) for some integer \(i\), and one writes \(G = \langle a \rangle\).<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> Cyclic groups are the simplest groups to describe, and two theorems settle their generators, their subgroups, and their isomorphism types.<sup>[2](https://notes.amittai.studio/abstract-algebra/subgroups-and-quotients/cyclic-groups)</sup>

| Key fact | Detail |
|---|---|
| Definition | \(G = \langle a \rangle\): every element is \(a^i\) for some integer \(i\)<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> |
| Isomorphism types | A finite cyclic group of order \(n\) is isomorphic to \(\mathbb{Z}/n\mathbb{Z}\); an infinite cyclic group is isomorphic to \((\mathbb{Z}, +)\)<sup>[3](https://math.mit.edu/research/highschool/primes/circle/documents/2022/Sophia%20%26%20Jaeyi.pdf)</sup> |
| Notation | \(\mathbb{Z}_n\), \(C_n\), or \(\mathbb{Z}/n\mathbb{Z}\) for the cyclic group of order \(n\)<sup>[4](https://mathworld.wolfram.com/CyclicGroup.html)</sup><sup> • </sup><sup>[5](https://maths.nuigalway.ie/~rquinlan/teaching/grouptheory.pdf)</sup> |
| Subgroups | Every subgroup of a cyclic group is cyclic; for each divisor \(d\) of \(|G|\) there is exactly one subgroup of order \(d\), generated by \(a^{|G|/d}\)<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> |
| Number of generators | A cyclic group of order \(n\) has \(\varphi(n)\) generators, since the unique subgroup of order \(d\) has \(\varphi(d)\) generators and \(\sum_{d \mid n} \varphi(d) = n\)<sup>[2](https://notes.amittai.studio/abstract-algebra/subgroups-and-quotients/cyclic-groups)</sup> |
| Order of the generator | The order of a cyclic group equals the order of its generator<sup>[6](https://mis.alagappauniversity.ac.in/siteAdmin/dde-admin/uploads/5/__UG_B.Sc._Mathematics_113%2051_Modern%20Algebra_BSc%20MATHS_5665.pdf)</sup> |
| Basic examples | \((\mathbb{Z}, +)\) is infinite cyclic with generator 1; the integers \(0, 1, \ldots, n-1\) form a cyclic group of order \(n\) under addition mod \(n\)<sup>[4](https://mathworld.wolfram.com/CyclicGroup.html)</sup><sup> • </sup><sup>[5](https://maths.nuigalway.ie/~rquinlan/teaching/grouptheory.pdf)</sup> |

## Definition and basic examples

A group \(G\) is cyclic when all of its members can be written as compositions of a single member with itself, that is \(G = \{e, a, a^2, a^3, \ldots\}\).<sup>[7](https://new.math.uiuc.edu/public402/Hvidsten.pdf)</sup> In additive notation the same idea reads \(G = \{0, a, 2a, 3a, \ldots\}\). An element \(g\) has finite order \(n\) if \(n\) is the smallest positive integer with \(g^n = e\), and infinite order otherwise.<sup>[8](https://www.math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>

The standard examples come from the integers. The integers \(0, 1, 2, \ldots, n-1\) form a cyclic group of order \(n\) under addition modulo \(n\), which is why the cyclic group of order \(n\) is denoted \(\mathbb{Z}_n\).<sup>[4](https://mathworld.wolfram.com/CyclicGroup.html)</sup><sup> • </sup><sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> The trivial group is a cyclic group of order 1.<sup>[8](https://www.math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>

**Classification.** These examples exhaust the possibilities: every cyclic group follows the shape of \(\mathbb{Z}/n\mathbb{Z}\), and it is infinite exactly in the case corresponding to \(n = 0\), where it looks like \(\mathbb{Z}\).<sup>[3](https://math.mit.edu/research/highschool/primes/circle/documents/2022/Sophia%20%26%20Jaeyi.pdf)</sup> A cyclic group of order \(n\) is commonly denoted \(C_n\).<sup>[5](https://maths.nuigalway.ie/~rquinlan/teaching/grouptheory.pdf)</sup> Some authors instead write \(\mathbb{Z}_n\), a notation that conflicts with number-theoretic uses such as \(\mathbb{Z}_p\) for the ring of \(p\)-adic numbers; to avoid this confusion, Bourbaki introduced the term *monogenous group* for a group with a single generator and restricted "cyclic group" to finite monogenous groups.<sup>[9](https://mathresearch.utsa.edu/wiki/index.php?title=Cyclic_Groups)</sup>

## Subgroups and the divisor structure

Cyclic groups have an elementary and complete subgroup theory. Every subgroup of a cyclic group is itself cyclic; moreover, when \(G = \langle a \rangle\) is a cyclic group, each divisor \(d\) of \(|G|\) corresponds to a single subgroup of order \(d\), namely the one produced by \(a^{|G|/d}\).<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> The proof of uniqueness shows that any subgroup's elements are powers of \(a^n\), where \(|G| = dn\), so the subgroup coincides with \(\langle a^n \rangle\).<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> In a finite cyclic group, each subgroup therefore has size dividing the size of the group.<sup>[10](https://www.isibang.ac.in/~athreya/swms2018f/cyclicgp.pdf)</sup>

The subgroup structure of a cyclic group of order \(n\) is thus the divisor structure of \(n\).<sup>[2](https://notes.amittai.studio/abstract-algebra/subgroups-and-quotients/cyclic-groups)</sup> Summing over the lattice of subgroups recovers the divisor identity \(\sum_{d \mid n} \varphi(d) = n\): every element of \(\mathbb{Z}/12\mathbb{Z}\), for instance, is a generator of exactly one subgroup.<sup>[2](https://notes.amittai.studio/abstract-algebra/subgroups-and-quotients/cyclic-groups)</sup>

Two consequences follow directly. Every group of composite order has proper subgroups: if \(\langle a \rangle \neq G\) for some \(a \neq e\) the claim is immediate, and otherwise \(\langle a^d \rangle\) is a proper subgroup for every divisor \(d\) of \(|G|\).<sup>[1](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)</sup> And in an infinite cyclic group the subgroups are exactly the groups \(d\mathbb{Z}\) of multiples of a fixed integer \(d\), with \(d\mathbb{Z} = (-d)\mathbb{Z}\).<sup>[11](https://terpconnect.umd.edu/~pbrosnan/notes/ugalg/sect0015.html)</sup>

## Generators

An element \(g^m\) of a cyclic group \(\langle g \rangle\) of order \(n\) generates the whole group precisely when its order is \(n\), and the count of generators is governed by [Euler's totient function](https://www.edgepedia.org/eulers-totient-function) \(\varphi\). The theorem follows because there is exactly one subgroup \(H\) of order \(d\) for each divisor \(d\) of \(n\), and \(H\) has \(\varphi(d)\) generators.<sup>[12](https://crypto.stanford.edu/pbc/notes/numbertheory/cyclic.html)</sup>

## Occurrences in mathematics

Cyclic groups appear wherever a single operation repeats. In geometry, the set of rotations in a finite symmetry group forms a cyclic group: a finite symmetry group with \(n\) rotations has its rotations generated by the rotation through \(360/n\) degrees, and any finite cyclic group \(C_n\) is realized as the rotational symmetry group of a regular \(n\)-gon.<sup>[7](https://new.math.uiuc.edu/public402/Hvidsten.pdf)</sup><sup> • </sup><sup>[13](https://berkeley.mt/archives/bmt-2014/PowS2014.pdf)</sup> In complex analysis and algebra, the group \(\mu_n\) of \(n\)-th roots of unity is cyclic with generator \(e^{2\pi i/n}\).<sup>[8](https://www.math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>

A deeper structural fact concerns fields: whenever a field F has a finite subgroup G lying inside the group of units formed by its non-zero elements under multiplication, that subgroup must be cyclic.<sup>[14](https://academic.oup.com/book/54486/chapter-abstract/422571419)</sup>

Cyclic groups also occupy a fixed place in the classification of finite simple groups, where one of the three infinite classes consists of the cyclic groups of prime order.<sup>[9](https://mathresearch.utsa.edu/wiki/index.php?title=Cyclic_Groups)</sup>

## Applications in cryptography

Because the discrete logarithm problem is posed in cyclic groups, they underpin public-key cryptography. The ElGamal cryptosystem, invented by Taher ElGamal in 1985, works over the finite cyclic group of the finite field.<sup>[15](https://arxiv.org/abs/1405.0914)</sup> Modern standards continue this line: NIST Special Publication 800-186 (2023) specifies the elliptic curves recommended for U.S. Government use in discrete-logarithm-based cryptography, including newly specified Edwards curves chosen for increased performance, side-channel resistance, and simpler implementation compared with traditional Weierstrass curves.<sup>[16](https://www.nist.gov/publications/recommendations-discrete-logarithm-based-cryptography-elliptic-curve-domain-parameters)</sup> Polycyclic groups, natural generalizations of cyclic groups with more complicated algorithmic properties, have been studied as a basis for cryptography resistant to quantum attack.<sup>[17](https://arxiv.org/abs/1607.05819)</sup>

## References

1. [Cyclic Groups, Stanford cryptography notes (Ben Lynn)](https://crypto.stanford.edu/pbc/notes/group/cyclic.html)
2. [Cyclic Groups, study notes by Amittai Axelrod](https://notes.amittai.studio/abstract-algebra/subgroups-and-quotients/cyclic-groups)
3. [Cyclic groups, MIT PRIMES Circle notes (2022)](https://math.mit.edu/research/highschool/primes/circle/documents/2022/Sophia%20%26%20Jaeyi.pdf)
4. [Cyclic Group, Wolfram MathWorld](https://mathworld.wolfram.com/CyclicGroup.html)
5. [Group Theory (MA343) Lecture Notes, National University of Ireland, Galway](https://maths.nuigalway.ie/~rquinlan/teaching/grouptheory.pdf)
6. [Modern Algebra, B.Sc. Mathematics course text, Alagappa University](https://mis.alagappauniversity.ac.in/siteAdmin/dde-admin/uploads/5/__UG_B.Sc._Mathematics_113%2051_Modern%20Algebra_BSc%20MATHS_5665.pdf)
7. [Symmetry, Chapter 6 of a geometry text, University of Illinois (M. Hvidsten)](https://new.math.uiuc.edu/public402/Hvidsten.pdf)
8. [Abstract Algebra, Chapter 2: Group theory, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/algebra-ch02.html)
9. [Cyclic Groups, UTSA Mathematics Research wiki](https://mathresearch.utsa.edu/wiki/index.php?title=Cyclic_Groups)
10. [Subgroups of Cyclic Groups, workshop notes, Indian Statistical Institute](https://www.isibang.ac.in/~athreya/swms2018f/cyclicgp.pdf)
11. [Subgroups of cyclic groups, UMD MATH 403 notes (P. Brosnan)](https://terpconnect.umd.edu/~pbrosnan/notes/ugalg/sect0015.html)
12. [Number Theory: Cyclic Groups, Stanford cryptography notes (Ben Lynn)](https://crypto.stanford.edu/pbc/notes/numbertheory/cyclic.html)
13. [Power of Symmetry, Berkeley Math Tournament 2014 materials](https://berkeley.mt/archives/bmt-2014/PowS2014.pdf)
14. [Finite cyclic groups and finite fields, An Introductory Course in Commutative Algebra, Oxford Academic](https://academic.oup.com/book/54486/chapter-abstract/422571419)
15. [The Discrete Logarithm problem in the ElGamal cryptosystem over U(n), arXiv:1405.0914](https://arxiv.org/abs/1405.0914)
16. [Recommendations for Discrete Logarithm-based Cryptography: Elliptic Curve Domain Parameters, NIST SP 800-186](https://www.nist.gov/publications/recommendations-discrete-logarithm-based-cryptography-elliptic-curve-domain-parameters)
17. [The Status of Polycyclic Group-Based Cryptography: A Survey and Open Problems, arXiv:1607.05819](https://arxiv.org/abs/1607.05819)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Group structures and subgroups*

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