Cyclic group

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In 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 a (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of a (or na when the notation is additive).

That is, we say G is cyclic if G = { an for any integer n }. 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 a is G itself suffices to show that G is cyclic.

For example, if G = { e, g1, g2, g3, g4, g5 }, then G is cyclic. And, G is essentially the same as (that is, isomorphic to) the group of { 0, 1, 2, 3, 4, 5 } for addition modulo 6. I.e. 1 + 2 mod 6 = 3, 2 + 5 mod 6 = 1, 4 - 4 = 4 + 2 mod 6 = 0 and so on. One can find an isomorphism by letting g = 1.

Up to isomorphism there exists exactly one cyclic group for every finite number of elements, and one infinite cyclic group. Hence, the cyclic groups are the simplest groups and they are completely classified.

Unlike the name suggests, it is possible to generate infinitely many elements and not form a literal cycle: that is, every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^n} is distinct. A group generated in this way is called an infinite cyclic group, every one of which is isomorphic to the additive group of integers Z.

Since the groups are Abelian they are often written additively, and denoted by Zn; however, this notation is often avoided by number theorists because it conflicts or is easily confused with the usual notation for p-adic number rings or localisation at a prime ideal. The quotient group notation Z/nZ (see also below) is an alternative.

One may write the group multiplicatively, and denote it by Cn. (For example, a3a4 = a2 in C5, whereas 3 + 4 = 2 (mod 5) in Z/5Z.)

Properties

Every cyclic group is isomorphic to (essentially the same as) the group { 0, 1, 2, ... n - 1 } under addition modulo n, or Z, the additive group of all of integers. Thus, one only needs to look at such groups to understand cyclic groups in general. This makes a cyclic group 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: ab = ba. This is so since a + b mod n = b + a mod n.
  • If n < Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^n = e} since n mod n = 0.
  • If n = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} , then there are exactly two generators: namely 1 and -1 for Z, and any others mapped to them under an isomorphism in other infinite cyclic groups.
  • 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.
  • Cn 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} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cong} { 0, 1, 2, 3, 4, ... n - 1} under addition modulo n.

The generators of Z/nZ are the residue classes of the integers which are coprime to n; the number of those generators is known as φ(n), where φ is Euler's totient function.

More generally, if d is a divisor of n, then the number of elements in Z/nZ 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.

The direct product of two cyclic groups Cn and Cm is cyclic if and only if n and m are coprime. Thus e.g. C12 is the direct product of C3 and C4, but not of C6 and C2.

The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many cyclic groups.

Zn and Z are also commutative rings. If p is a prime, Zp is a finite field, also denoted by Fp or GF(p). Every other field with p elements is isomorphic to this one.

The units of the ring Zn are the numbers coprime to n. They form a group under multiplication modulo n; it has φ(n) elements (see above). It is written as Zn×.

For example, for n = 6 we get {1,5}, and for n = 8 we get {1,3,5,7}.

It is cyclic if and only if n is 2 or 4 or pk or 2 pk for an odd prime number p and k ≥ 1.

Thus for n = 6 it is cyclic, but not for n=8: that is isomorphic to the Klein four-group.

The generators of this cyclic group are called primitive roots modulo n.

The group Zp× is cyclic with p -1 elements for every prime p. 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 cyclic symmetry groups in 3D.

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., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = z^3 - 1 = (z - s^0)(z - s^1)(z - s^2)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = e^{2 \pi i /3}} and a group of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ s^0, s^1, s^2 \}} 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.

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8

Subgroups

All subgroups and factor groups of cyclic groups are cyclic. Specifically, the subgroups of Z are of the form mZ, with m an integer ≥0. All 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}. For every positive divisor d of n, the 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.

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 m.k is divisible by n. It is therefore n/a where a = (k, n) is the gcd of k and n. Put another way, the index of the subgroup generated by gk is a. This reasoning is known as the index calculus, in number theory.

Endomorphisms

The endomorphism ring of the abelian group Zn is isomorphic to itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Zn which maps each element to the sum of r copies of it. This is a bijection iff r is coprime with n, so the automorphism group of Zn is isomorphic to the group Zn× (see above). The automorphism group of Zn is sometimes called the character group of Zn and the construction of this group leads directly to the definition of Dirichlet characters.

Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z, and its automorphism group is isomorphic to the group of units of the ring Z, i.e. to {−1, +1} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cong} Z2.

See also

cs:Cyklická grupa de:Zyklische Gruppe fr:Groupe cyclique ko:순환군 he:חבורה ציקלית hu:Ciklikus csoport nl:Cyclische groep sv:Cyklisk grupp