Algebraic structure → Group theoryGroup theory |
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In group theory, a branch of abstract algebra, a **cyclic group** or **monogenous group** is a group that is generated by a single element.^{ [1] } That is, it is a set of invertible elements with a single associative binary operation, and it contains an element *g* such that every other element of the group may be obtained by repeatedly applying the group operation to *g* or its inverse. Each element can be written as a power of *g* in multiplicative notation, or as a multiple of *g* in additive notation. This element *g* is called a * generator * of the group.^{ [1] }

- Definition and notation
- Examples
- Integer and modular addition
- Modular multiplication
- Rotational symmetries
- Galois theory
- Subgroups
- Additional properties
- Associated objects
- Representations
- Cycle graph
- Cayley graph
- Endomorphisms
- Tensor product and Hom of cyclic groups
- Related classes of groups
- Virtually cyclic groups
- Locally cyclic groups
- Cyclically ordered groups
- Metacyclic and polycyclic groups
- See also
- Footnotes
- Notes
- Citations
- References
- Further reading
- External links

Every infinite cyclic group is isomorphic to the additive group of **Z**, the integers. Every finite cyclic group of order *n* is isomorphic to the additive group of **Z**/*n***Z**, the integers modulo *n*. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.

Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.

For any element *g* in any group *G*, one can form the subgroup of all integer powers ⟨*g*⟩ = {*g*^{k}|*k* ∈ **Z**}, called the **cyclic subgroup** of *g*. The order of *g* is the number of elements in ⟨*g*⟩; that is, the order of an element is equal to the order of its cyclic subgroup.

A *cyclic group* is a group which is equal to one of its cyclic subgroups: *G* = ⟨*g*⟩ for some element *g*, called a *generator*.

For a **finite cyclic group***G* of order *n* we have *G* = {*e*, *g*, *g*^{2}, ... , *g*^{n−1}}, where *e* is the identity element and *g*^{i} = *g*^{j} whenever *i* ≡ *j* (mod *n*); in particular *g*^{n} = *g*^{0} = *e*, and *g*^{−1} = *g*^{n−1}. An abstract group defined by this multiplication is often denoted C* _{n},* and we say that

For example, the set of complex 6th roots of unity

forms a group under multiplication. It is cyclic, since it is generated by the primitive root that is, *G* = ⟨*z*⟩ = { 1, *z*, *z*^{2}, *z*^{3}, *z*^{4}, *z*^{5} } with *z*^{6} = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C_{6} = ⟨*g*⟩ = { *e*, *g*, *g*^{2}, *g*^{3}, *g*^{4}, *g*^{5} } with multiplication *g ^{j}* ·

Instead of the quotient notations **Z**/*n***Z**, **Z**/(*n*), or **Z**/*n*, some authors denote a finite cyclic group as **Z*** _{n}*, but this conflicts with the notation of number theory, where

p1, (*∞∞) | p11g, (22∞) |
---|---|

| |

Two frieze groups are isomorphic to Z. With one generator, p1 has translations and p11g has glide reflections. |

On the other hand, in an **infinite cyclic group***G =* ⟨*g*⟩*,* the powers *g*^{k} give distinct elements for all integers *k*, so that *G* = { ... , *g*^{−2}, *g*^{−1}, *e*, *g*, *g*^{2}, ... }, and *G* is isomorphic to the standard group C = C_{∞} and to **Z**, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the name "cyclic" may be misleading.^{ [2] }

To avoid this confusion, Bourbaki introduced the term **monogenous group** for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".^{ [note 1] }

C_{1} | C_{2} | C_{3} |
---|---|---|

C_{4} | C_{5} | C_{6} |

The set of integers **Z**, with the operation of addition, forms a group.^{ [1] } It is an **infinite cyclic group**, because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to **Z**.

For every positive integer *n*, the set of integers modulo *n*, again with the operation of addition, forms a finite cyclic group, denoted **Z**/*n***Z**.^{ [1] } A modular integer *i* is a generator of this group if *i* is relatively prime to *n*, because these elements can generate all other elements of the group through integer addition. (The number of such generators is *φ*(*n*), where *φ* is the Euler totient function.) Every finite cyclic group *G* is isomorphic to **Z**/*n***Z**, where *n* = |*G*| is the order of the group.

The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings, also denoted **Z** and **Z**/*n***Z** or **Z**/(*n*). If *p* is a prime, then **Z**/*p Z* is a finite field, and is usually denoted

For every positive integer *n*, the set of the integers modulo *n* that are relatively prime to *n* is written as (**Z**/*n***Z**)^{×}; it forms a group under the operation of multiplication. This group is not always cyclic, but is so whenever *n* is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime (sequence A033948 in the OEIS ).^{ [4] }^{ [5] } This is the multiplicative group of units of the ring **Z**/*n***Z**; there are *φ*(*n*) of them, where again *φ* is the Euler totient function. For example, (**Z**/6**Z**)^{×} = {1,5}, and since 6 is twice an odd prime this is a cyclic group. In contrast, (**Z**/8**Z**)^{×} = {1,3,5,7} is a Klein 4-group and is not cyclic. When (**Z**/*n***Z**)^{×} is cyclic, its generators are called primitive roots modulo *n*.

For a prime number *p*, the group (**Z**/*p***Z**)^{×} is always cyclic, consisting of the non-zero elements of the finite field of order *p*. More generally, every finite subgroup of the multiplicative group of any field is cyclic.^{ [6] }

The set of rotational symmetries of a polygon forms a finite cyclic group.^{ [7] } If there are *n* different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to **Z**/*n***Z**. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead rotoreflections.

The group of all rotations of a circle *S*^{1} (the circle group, also denoted *S*^{1}) is *not* cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C_{∞} is countable, while *S*^{1} is not. The group of rotations by rational angles *is* countable, but still not cyclic.

An *n*th root of unity is a complex number whose *n*th power is 1, a root of the polynomial *x*^{n} − 1. The set of all *n*th roots of unity form a cyclic group of order *n* under multiplication.^{ [1] } For example, the polynomial *z*^{3} − 1 factors as (*z* − 1)(*z* − *ω*)(*z* − *ω*^{2}), where *ω* = *e*^{2πi/3}; the set {1, *ω*, *ω*^{2}} = {*ω*^{0}, *ω*^{1}, *ω*^{2}} forms a cyclic group under multiplication. The Galois group of the field extension of the rational numbers generated by the *n*th roots of unity forms a different group, isomorphic to the multiplicative group (**Z/***n***Z**)^{×} of order *φ*(*n*), which is cyclic for some but not all *n* (see above).

A field extension is called a cyclic extension if its Galois group is cyclic. For fields of characteristic zero, such extensions are the subject of Kummer theory, and are intimately related to solvability by radicals. For an extension of finite fields of characteristic *p*, its Galois group is always finite and cyclic, generated by a power of the Frobenius mapping.^{ [8] } Conversely, given a finite field *F* and a finite cyclic group *G*, there is a finite field extension of *F* whose Galois group is *G*.^{ [9] }

All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of **Z** are of the form ⟨*m*⟩ = *m***Z**, with *m* a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0**Z**, they all are isomorphic to **Z**. The lattice of subgroups of **Z** is isomorphic to the dual of the lattice of natural numbers ordered by divisibility.^{ [10] } Thus, since a prime number *p* has no nontrivial divisors, *p***Z** is a maximal proper subgroup, and the quotient group **Z**/*p***Z** is simple; in fact, a cyclic group is simple if and only if its order is prime.^{ [11] }

All quotient groups **Z**/*n***Z** are finite, with the exception **Z**/0**Z** = **Z**/{0}. For every positive divisor *d* of *n*, the quotient group **Z**/*n***Z** has precisely one subgroup of order *d*, generated by the residue class of *n*/*d*. There are no other subgroups.

Every cyclic group is abelian.^{ [1] } That is, its group operation is commutative: *gh* = *hg* (for all *g* and *h* in *G*). This is clear for the groups of integer and modular addition since *r* + *s* ≡ *s* + *r* (mod *n*), and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order *n*, *g*^{n} is the identity element for any element *g*. This again follows by using the isomorphism to modular addition, since *kn* ≡ 0 (mod *n*) for every integer *k*. (This is also true for a general group of order *n*, due to Lagrange's theorem.)

For a prime power *p ^{k},* the group

Because a cyclic group is abelian, each of its conjugacy classes consists of a single element. A cyclic group of order *n* therefore has *n* conjugacy classes.

If *d* is a divisor of *n*, then the number of elements in **Z**/*n***Z** which have order *d* is *φ*(*d*), and the number of elements whose order divides *d* is exactly *d*. If *G* is a finite group in which, for each *n* > 0, *G* contains at most *n* elements of order dividing *n*, then *G* must be cyclic.^{ [note 2] } The order of an element *m* in **Z**/*n***Z** is *n*/gcd(*n*,*m*).

If *n* and *m* are coprime, then the direct product of two cyclic groups **Z**/*n***Z** and **Z**/*m***Z** is isomorphic to the cyclic group **Z**/*nm***Z**, and the converse also holds: this is one form of the Chinese remainder theorem. For example, **Z**/12**Z** is isomorphic to the direct product **Z**/3**Z** × **Z**/4**Z** under the isomorphism (*k* mod 12) → (*k* mod 3, *k* mod 4); but it is not isomorphic to **Z**/6**Z** × **Z**/2**Z**, in which every element has order at most 6.

If *p* is a prime number, then any group with *p* elements is isomorphic to the simple group **Z**/*p***Z**. A number *n* is called a cyclic number if **Z**/*n***Z** is the only group of order *n*, which is true exactly when gcd(*n*,*φ*(*n*)) = 1.^{ [13] } The cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are:

- 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ... (sequence A003277 in the OEIS )

The definition immediately implies that cyclic groups have group presentation C_{∞} = ⟨*x* | ⟩ and C_{n} = ⟨*x* | *x*^{n}⟩ for finite *n*.^{ [14] }

The representation theory of the cyclic group is a critical base case for the representation theory of more general finite groups. In the complex case, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent. In the positive characteristic case, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocks of cyclic defect.

A **cycle graph** illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. Trivial paths (identity) can be drawn as a loop but are usually suppressed. Z_{2} is sometimes drawn with two curved edges as a multigraph.^{ [15] }

A cyclic groups Z_{n}, with order *n*, corresponds to a single cycle graphed simply as an *n*-sided polygon with the elements at the vertices.

A Cayley graph is a graph defined from a pair (*G*,*S*) where *G* is a group and *S* is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs.^{ [16] } These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the vertex-transitive graphs whose symmetry group includes a transitive cyclic group.^{ [17] }

The endomorphism ring of the abelian group **Z**/*n***Z** is isomorphic to **Z**/*n***Z** itself as a ring.^{ [18] } Under this isomorphism, the number *r* corresponds to the endomorphism of **Z**/*n***Z** that maps each element to the sum of *r* copies of it. This is a bijection if and only if *r* is coprime with *n*, so the automorphism group of **Z**/*n***Z** is isomorphic to the unit group (**Z**/*n***Z**)^{×}.^{ [18] }

Similarly, the endomorphism ring of the additive group of **Z** is isomorphic to the ring **Z**. Its automorphism group is isomorphic to the group of units of the ring **Z**, which is ({−1, +1}, ×) ≅ C_{2}.

The tensor product **Z**/*m***Z** ⊗ **Z**/*n***Z** can be shown to be isomorphic to **Z** / gcd(*m*, *n*)**Z**. So we can form the collection of group homomorphisms from **Z**/*m***Z** to **Z**/*n***Z**, denoted hom(**Z**/*m***Z**, **Z**/*n***Z**), which is itself a group.

For the tensor product, this is a consequence of the general fact that *R*/*I* ⊗_{R}*R*/*J* ≅ *R*/(*I* + *J*), where *R* is a commutative ring with unit and *I* and *J* are ideals of the ring. For the Hom group, recall that it is isomorphic to the subgroup of **Z** / *n***Z** consisting of the elements of order dividing *m*. That subgroup is cyclic of order gcd(*m*, *n*), which completes the proof.

Several other classes of groups have been defined by their relation to the cyclic groups:

A group is called **virtually cyclic** if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends;^{ [note 3] } an example of such a group is the direct product of **Z**/*n***Z** and **Z**, in which the factor **Z** has finite index *n*. Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.^{ [20] }

A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice.^{ [21] }

A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group). Every finite subgroup of a cyclically ordered group is cyclic.^{ [22] }

A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic.^{ [23] } These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a cyclic quotient, ending in the trivial group. Every finitely generated abelian group or nilpotent group is polycyclic.^{ [24] }

- Cycle graph (group)
- Cyclic module
- Cyclic sieving
- Prüfer group (countably infinite analogue)
- Circle group (uncountably infinite analogue)

- ↑ DEFINITION 15.
*A group is called*monogenous*if it admits a system of generators consisting of a single element. A finite monogenous group is called*cyclic.^{ [3] } - ↑ This implication remains true even if only prime values of
*n*are considered.^{ [12] }(And observe that when*n*is prime, there is exactly one element whose order is a proper divisor of*n*, namely the identity.) - ↑ If
*G*has two ends, the explicit structure of*G*is well known:*G*is an extension of a finite group by either the infinite cyclic group or the infinite dihedral group.^{ [19] }

- 1 2 3 4 5 6 "Cyclic group",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ ( Lajoie & Mura 2000 , pp. 29–33).
- (Bourbaki 1998, p. 49) or
*Algebra I: Chapters 1–3*, p. 49, at Google Books. - ↑ ( Motwani & Raghavan 1995 , p. 401).
- ↑ ( Vinogradov 2003 , pp. 105–132, § VI PRIMITIVE ROOTS AND INDICES).
- ↑ ( Rotman 1998 , p. 65).
- ↑ ( Stewart & Golubitsky 2010 , pp. 47–48).
- ↑ ( Cox 2012 , p. 294, Theorem 11.1.7).
- ↑ ( Cox 2012 , p. 295, Corollary 11.1.8 and Theorem 11.1.9).
- ↑ ( Aluffi 2009 , pp. 82–84, 6.4 Example: Subgroups of Cyclic Groups).
- ↑ ( Gannon 2006 , p. 18).
- (Gallian 2010, p. 84, Exercise 43).
- ↑ ( Jungnickel 1992 , pp. 545–547).
- ↑ ( Coxeter & Moser 1980 , p. 1).
- ↑ Weisstein, Eric W. "Cycle Graph".
*MathWorld*. - ↑ ( Alspach 1997 , pp. 1–22).
- ↑ ( Vilfred 2004 , pp. 34–36).
- 1 2 ( Kurzweil & Stellmacher 2004 , p. 50).
- (Stallings 1970, pp. 124–128). See in particular
*Groups of cohomological dimension one*, p. 126, at Google Books. - ↑ ( Alonso 1991 , Corollary 3.6).
- ↑ ( Ore 1938 , pp. 247–269).
- ↑ ( Fuchs 2011 , p. 63).
- ↑ A. L. Shmel'kin (2001) [1994], "Metacyclic group",
*Encyclopedia of Mathematics*, EMS Press - ↑ "Polycyclic group",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, a **finite field** or **Galois field** is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod *p* when *p* is a prime number.

In mathematics, a **group** is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

In mathematics, specifically group theory, given a prime number *p*, a ** p-group** is a group in which the order of every element is a power of

In mathematics, a **simple group** is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

In mathematics, the **free group***F*_{S} over a given set *S* consists of all words that can be built from members of *S*, considering two words to be different unless their equality follows from the group axioms. The members of *S* are called **generators** of *F*_{S}, and the number of generators is the **rank** of the free group. An arbitrary group *G* is called **free** if it is isomorphic to *F*_{S} for some subset *S* of *G*, that is, if there is a subset *S* of *G* such that every element of *G* can be written in exactly one way as a product of finitely many elements of *S* and their inverses.

In abstract algebra, a **generating set of a group** is a subset of the group set such that every element of the group can be expressed as a combination of finitely many elements of the subset and their inverses.

In mathematics, more specifically in the field of group theory, a **solvable group** or **soluble group** is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

In number theory, the **ideal class group** of an algebraic number field *K* is the quotient group *J _{K}*/

In mathematics, a **free abelian group** or **free Z-module** is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In abstract algebra, a **finite group** is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.

In mathematics, a **Cayley graph**, also known as a **Cayley colour graph**, **Cayley diagram**, **group diagram**, or **colour group** is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.

In abstract algebra, a module is **indecomposable** if it is non-zero and cannot be written as a direct sum of two non-zero submodules.

In algebra, a **finitely generated group** is a group *G* that has some finite generating set *S* so that every element of *G* can be written as the combination of finitely many elements of the finite set *S* and of inverses of such elements.

In group theory, a subfield of abstract algebra, a group **cycle graph** illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.

In mathematics, specifically in group theory, an **elementary abelian group** is an abelian group in which every nontrivial element has order *p*. The number *p* must be prime, and the elementary abelian groups are a particular kind of *p*-group. The case where *p* = 2, i.e., an elementary abelian 2-group, is sometimes called a **Boolean group**.

In mathematics, specifically in group theory, the **Prüfer p-group** or the

In mathematics, a **class formation** is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.

In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order *n*, every subgroup's order is a divisor of *n*, and there is exactly one subgroup for each divisor. This result has been called the **fundamental theorem of cyclic groups**.

- Alonso, J. M.; et al. (1991), "Notes on word hyperbolic groups",
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*Generators and Relations for Discrete Groups*, New York: Springer-Verlag, p. 1, ISBN 0-387-09212-9 - Lajoie, Caroline; Mura, Roberta (November 2000), "What's in a name? A learning difficulty in connection with cyclic groups",
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*Galois Theory*, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, Theorem 11.1.7, p. 294, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9 - Gallian, Joseph (2010),
*Contemporary Abstract Algebra*(7th ed.), Cengage Learning, Exercise 43, p. 84, ISBN 978-0-547-16509-7 - Gannon, Terry (2006),
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