Cyclic group

Group theory
Group theory
Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product
Finite groups (classification) Cyclic group Zn Symmetric group, Sn Dihedral group, Dn Alternating group An Mathieu groups M11, M12, M22, M23, M24 Conway groups Co1, Co2, Co3 Janko groups J1, J2, J3, J4 Fischer groups F22, F23, F24 Baby Monster group B Monster group M
Discrete groups and lattices The integers, Z Lattice (group) Modular groups, PSL(2,Z) and SL(2,Z)
Topological groups and Lie groups Solenoid (mathematics) Circle group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G2 F4 E6 E7 E8 Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)
v · group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive). Definition The six 6th complex roots of unity form a cyclic group under multiplication. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2. A group G is called cyclic if there exists an element g in G such that G = <g> = { gn | n is an integer }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains g is G itself suffices to show that G is cyclic. For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6. For example, 1 + 2 = 3 (mod 6) corresponds to g1·g2 = g3, and 2 + 5 = 1 (mod 6) corresponds to g2·g5 = g7 = g1, and so on. One can use the isomorphism φ defined by φ(gi) = i. For every positive integer n there is exactly one cyclic group (up to isomorphism) whose order is n, and there is exactly one infinite cyclic group (the integers under addition). Hence, the cyclic groups are the simplest groups and they are completely classified. The name "cyclic" may be misleading: it is possible to generate infinitely many elements and not form any literal cycles; that is, every gn is distinct. (It can be said that it has one infinitely long cycle.) A group generated in this way is called an infinite cyclic group, and is isomorphic to the additive group of integers Z. Furthermore, the circle group (whose elements are uncountable) is not a cyclic group—a cyclic group always has countable elements. Since the cyclic groups are abelian, they are often written additively and denoted Zn. However, this notation can be problematic for number theorists because it conflicts with the usual notation for p-adic number rings or localization at a prime ideal. The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. We adopt the first of these here to avoid the collision of notation. See also the section Subgroups and notation below. One may write the group multiplicatively, and denote it by Cn, where n is the order (which can be ∞). For example, g3g4 = g2 in C5, whereas 3 + 4 = 2 in Z/5Z. Properties Cycle graph of the cyclic group Z15 The elements of the subroups Z3 and Z5 are indicated by the trigon and the pentagon. The fundamental theorem of cyclic groups states that if G is a cyclic group of order n then every subgroup of G is cyclic. Moreover, the order of any subgroup of G is a divisor of n and for each positive divisor k of n the group G has exactly one subgroup of order k. This property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor d of n the group has at most one subgroup of order d. Sometimes the refined statement is used: a group of order n is cyclic if and only if for every divisor d of n the group has exactly one subgroup of order d. Every finite cyclic group is isomorphic to the group { [0], [1], [2], ..., [n − 1] } of integers modulo n under addition, and any infinite cyclic group is isomorphic to Z (the set of all integers) under addition. Thus, one only needs to look at such groups to understand the properties of cyclic groups in general. Hence, cyclic groups are one of the simplest groups to study and a number of nice properties are known. Given a cyclic group G of order n (n may be infinity) and for every g in G, G is abelian; that is, their group operation is commutative: gh = hg (for all h in G). This is so since g + h mod n = h + g mod n. If n is finite, then gn = g0 is the identity element of the group, since kn mod n = 0 for any integer k. If n = ∞, then there are exactly two elements that generate the group on their own: namely 1 and −1 for Z If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient function Every subgroup of G is cyclic. Indeed, each finite subgroup of G is a group of { 0, 1, 2, 3, ... m − 1} with addition modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to (so isomorphic to) Z. Gn is isomorphic to Z/nZ (factor group of Z over nZ) since Z/nZ = {0 + nZ, 1 + nZ, 2 + nZ, 3 + nZ, 4 + nZ, ..., n − 1 + nZ} $\cong$ { 0, 1, 2, 3, 4, ..., n − 1} under addition modulo n. More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). The order of the residue class of m is n / gcd(n,m). If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group Cp or Z/pZ. There are more numbers with the same property, see cyclic number. The direct product of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime. Thus e.g. Z/12Z is the direct product of Z/3Z and Z/4Z, but not the direct product of Z/6Z and Z/2Z. The definition immediately implies that cyclic groups have very simple group presentation C∞ = < x | > and Cn = < x | xn > for finite n. A primary cyclic group is a group of the form Z/pkZ where p is a prime number. The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups. Z/nZ and Z are also commutative rings. If p is a prime, then Z/pZ is a finite field, also denoted by Fp or GF(p). Every field with p elements is isomorphic to this one. The units of the ring Z/nZ are the numbers coprime to n. They form a group under multiplication modulo n with φ(n) elements (see above). It is written as (Z/nZ)×. For example, when n = 6, we get (Z/nZ)× = {1,5}. When n = 8, we get (Z/nZ)× = {1,3,5,7}. In fact, it is known that (Z/nZ)× is cyclic if and only if n is 1 or 2 or 4 or pk or 2 pk for an odd prime number p and k ≥ 1, in which case every generator of (Z/nZ)× is called a primitive root modulo n. Thus, (Z/nZ)× is cyclic for n = 6, but not for n = 8, where it is instead isomorphic to the Klein four-group. The group (Z/pZ)× is cyclic with p − 1 elements for every prime p, and is also written (Z/pZ)* because it consists of the non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic. Examples In 2D and 3D the symmetry group for n-fold rotational symmetry is Cn, of abstract group type Zn. In 3D there are also other symmetry groups which are algebraically the same, see Symmetry groups in 3D that are cyclic as abstract group. Note that the group S1 of all rotations of a circle (the circle group) is not cyclic, since it is not even countable. The nth roots of unity form a cyclic group of order n under multiplication. e.g., 0 = z3 − 1 = (z − s0)(z − s1)(z − s2) where si = e2πi / 3 and a group of {s0,s1,s2} under multiplication is cyclic. The Galois group of every finite field extension of a finite field is finite and cyclic; 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. Representation The cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices. The dark vertex in the cycle graphs below stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element. C1 C2 C3 C4 C5 C6 C7 C8 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. Subgroups and notation All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form mZ, with m an integer ≥0. All of these subgroups are different, and apart from the trivial group (for m=0) 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. All factor groups of Z are finite, except for the trivial exception Z/{0} = Z/0Z. For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, the one generated by the residue class of n/d. There are no other subgroups. The lattice of subgroups is thus isomorphic to the set of divisors of n, ordered by divisibility. In particular, a cyclic group is simple if and only if its order (the number of its elements) is prime.[1] Using the quotient group formalism, Z/nZ is a standard notation for the additive cyclic group with n elements. In ring terminology, the subgroup nZ is also the ideal (n), so the quotient can also be written Z/(n) or Z/n without abuse of notation. These alternatives do not conflict with the notation for the p-adic integers. The last form is very common in informal calculations; it has the additional advantage that it reads the same way that the group or ring is often described verbally in English, "Zee mod en". As a practical problem, one may be given a finite subgroup C of order n, generated by an element g, and asked to find the size m of the subgroup generated by gk for some integer k. Here m will be the smallest integer > 0 such that mk is divisible by n. It is therefore n/m where m = (k, n) is the greatest common divisor of k and n. Put another way, the index of the subgroup generated by gk is m. This reasoning is known as the index calculus algorithm, in number theory. Endomorphisms The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z/nZ 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/nZ is isomorphic to the unit group (Z/nZ)× (see above). Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, i.e. to {−1, +1} $\cong$ C2. Virtually 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 applying a member of the cyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. It is known that a finitely generated discrete group with exactly two ends is virtually cyclic (for instance the product of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic. See also Cyclic extension Cyclic module Cyclically ordered group Locally cyclic group, a group in which each finitely generated subgroup is cyclic Modular arithmetic Notes ^ Gannon (2006), p. 18 References Gallian, Joseph (1998), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin, ISBN 978-0-669-86179-2 , especially chapter 4. Herstein, I. N. (1996), Abstract algebra (3rd ed.), Prentice Hall, ISBN 978-0-13-374562-7, MR1375019 , especially pages 53–60. Gannon, Terry (2006), Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics, Cambridge monographs on mathematical physics, Cambridge University Press, ISBN 9780521835312  External links An introduction to cyclic groups Further reading Lajoie, Caroline; Mura, Roberta (November 2000), "What's in a Name? A Learning Difficulty in Connection with Cyclic Groups", For the Learning of Mathematics 20 (3): 29–33, JSTOR 40248334  Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Non-Departmental Public Body Stock market Look at other dictionaries: cyclic group — noun (mathematics) A group on which every element can be expressed as a power of a single element • • • Main Entry: ↑cycle …   Useful english dictionary cyclic group — ciklinė grupė statusas T sritis fizika atitikmenys: angl. cyclic group vok. zyklische Gruppe, f rus. циклическая группа, f pranc. groupe cyclique, m …   Fizikos terminų žodynas Locally cyclic group — In group theory, a locally cyclic group is a group ( G , *) in which every finitely generated subgroup is cyclic.ome facts*Every cyclic group is locally cyclic, and every locally cyclic group is abelian. *Every finitely generated locally cyclic… …   Wikipedia Free-by-cyclic group — In group theory, a group G is said to be free by cyclic if it has a free normal subgroup F such that the quotient group : G/F is cyclic. In other words, G is free by cyclic if it can be expressed as a group extension of a free group by a cyclic… …   Wikipedia Binary cyclic group — In mathematics, the binary cyclic group of the n gon is the cyclic group of order 2 n , C {2n}, thought of as a extension of the cyclic group C n a cyclic group of order 2.It is the binary polyhedral group corresponding to the cyclic group.ee… …   Wikipedia Primary cyclic group — A primary cyclic group is a cyclic group of prime power order:C {p^m};(for any prime p , and natural number m ).Every finite abelian group G may be written as a finite direct sum of primary cyclic groups::G=igoplus {1leq i leq n}C p i}^{m i;This …   Wikipedia Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia Cyclic order — In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order cannot be modeled as a binary relation a < b . One does not say that east is more clockwise than west.… …   Wikipedia Cyclic (mathematics) — There are many terms in mathematics that begin with cyclic: Cyclic chain rule, for derivatives, used in thermodynamics Cyclic code, linear codes closed under cyclic permutations Cyclic convolution, a method of combining periodic functions Cycle… …   Wikipedia Cyclic module — In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element. 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