# Cyclical group

In the group theory , a is cyclic group , a group derived from a single element produced is. It consists only of the potencies of the producer : ${\ displaystyle a}$ ${\ displaystyle a}$

${\ displaystyle \ left \ langle a \ right \ rangle: = \ lbrace a ^ {n} \ mid n \ in \ mathbb {Z} \ rbrace.}$

So a group is cyclic if it contains an element such that each element is a power of . This means that there is an element such that itself is the only subgroup of that contains. In this case, a generating element or, for short, a generator of is called. ${\ displaystyle G}$${\ displaystyle a}$${\ displaystyle G}$${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle G}$

Cyclic groups are the simplest groups and can be fully classified: for every natural number (for this statement we do not consider 0 a natural number) there is a cyclic group with exactly elements, and there is the infinite cyclic group , the additive group of the whole Numbers . Every other cyclic group is isomorphic to one of these groups . ${\ displaystyle n}$${\ displaystyle C_ {n}}$${\ displaystyle n}$ ${\ displaystyle \ mathbb {Z}}$

## illustration

### Turning groups

The finite cyclic groups can be illustrated as rotating groups of regular polygons in the plane. For example, the group consists of the possible rotations of the plane that transform a given square into itself. ${\ displaystyle C_ {4}}$

The figure above shows a square A and the positions B, C and D, into which it can be converted by turning. The rotation required for this is indicated below. The elements of the cyclic group here are the movements and not the positions of the square. This means that the group in this representation consists of the set {0 °, 90 °, 180 °, 270 °}. The linking of the elements is the sequential execution of the rotations; this corresponds to an addition of the angles. The rotation by 360 ° corresponds to the rotation by 0 °, i.e. the angles are, strictly speaking, added modulo 360 °. ${\ displaystyle C_ {4}}$

If not only rotations of the plane are allowed, but also reflections, then in the case of polygons the so-called dihedral groups are obtained .

The rotation group of the circle , is not cyclical. ${\ displaystyle S ^ {1}}$

### Residual class groups

Another representation of a cyclic group is the addition modulo of a number, the so-called remainder class arithmetic . In the additive group , the remainder class of 1 is a producer, that is, you can get any other remainder class by repeatedly adding the 1 to itself. Using the example , this means that all 4 elements can be represented as the sum of 1, i.e. 1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, 0 = 1 + 1 + 1 + 1. The remainder class group behaves exactly like the rotation group described above {0 °, 90 °, 180 °, 270 °}: 0 corresponds to 0 °, 1 corresponds to 90 ° etc .: These two groups are isomorphic . ${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}, +)}$${\ displaystyle \ mathbb {Z} / 4 \ mathbb {Z} = \ {0,1,2,3 \}}$${\ displaystyle \ mathbb {Z} / 4 \ mathbb {Z}}$

## Notations

For the finite cyclic groups the three notations are essentially used: , and . For the non-finite cyclic group there are the notations and . As a group operation, addition is referred to in , and . In the group operation is often written multiplicative. ${\ displaystyle C_ {n}}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {n}}$${\ displaystyle C _ {\ infty}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {n}}$${\ displaystyle C_ {n}}$

The designations , and come from the fact that the additive groups of the remainder class rings are by themselves the best-known representatives of cyclic groups. All of these structures are even rings which, in addition to the addition relevant here, also have a multiplicative link (not used here). ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {n}}$${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$

The term is also used for the n -adic numbers . ${\ displaystyle \ mathbb {Z} _ {n}}$

## properties

All cyclic groups are Abelian groups .

A cyclic group can have several producers. The producers of are +1 and −1, the producers of are the remainder classes, which are coprime too ; their number is given by Euler's φ function . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle n}$${\ displaystyle \ varphi (n)}$

Is in general a factor of , then the number of elements of that have the order : ${\ displaystyle d}$${\ displaystyle n}$${\ displaystyle \ varphi (d)}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$ ${\ displaystyle d}$

${\ displaystyle {\ Big |} \ {m \ in \ mathbb {Z} / n \ mathbb {Z} \ mid {\ text {ord}} (m) = d \} {\ Big |} = \ varphi ( d)}$.

The direct product of two cyclic groups and is cyclic if and only if and are coprime ; in this case the product is isomorphic to . ${\ displaystyle C_ {n}}$${\ displaystyle C_ {m}}$${\ displaystyle n}$${\ displaystyle m}$ ${\ displaystyle C_ {mn}}$

Every finitely generated Abelian group is a direct product of finitely many (finite and infinite) cyclic groups.

The group exponent of a finite cyclic group is equal to its order . Every finite cyclic group is isomorphic to the additive group of the remainder class ring , the isomorphism is the discrete logarithm : If is a generator of , then the mapping is ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle a}$${\ displaystyle C_ {n}}$

${\ displaystyle a ^ {t} \ mapsto t}$ mod ${\ displaystyle n}$

an isomorphism .

### Subgroups and Factor Groups

All subgroups and factor groups of cyclic groups are cyclic. In particular, the subgroups of are of the form with a natural number . For various these subgroups are different, and for they are isomorphic to . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle m \ mathbb {Z}}$${\ displaystyle m \ in \ mathbb {N} _ {0}}$${\ displaystyle m}$${\ displaystyle m \ not = 0}$${\ displaystyle \ mathbb {Z}}$

The association of the subgroups of is isomorphic to the dual association of natural numbers with divisibility . All factor groups of are finite, with the exception of the trivial factor group . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / \ {0 \}}$

For every positive factor of the group has exactly one subgroup of the order , namely the subgroup generated by the element . There are no other than these subgroups. The subgroup lattice is therefore isomorphic to the divisor lattice of . ${\ displaystyle d}$${\ displaystyle n}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$ ${\ displaystyle d}$${\ displaystyle n / d}$${\ displaystyle \ left \ {kn / d \ mid k = 0, \ ldots, d-1 \ right \}}$${\ displaystyle n}$

A cyclic group is simple if and only if its order is a prime number.

### Endomorphisms and automorphisms

The endomorphism ring (see group homomorphism ) of the group is ring isomorphic to the remainder class ring . Under this isomorphism the residue class corresponds of the endomorphism of which each element to its mapping th power. It follows that the automorphism of isomorphic to the group , the device group of the ring is. This group consists of the elements that are coprime and thus has exactly elements. ${\ displaystyle C_ {n}}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle r}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle C_ {n}}$${\ displaystyle r}$${\ displaystyle C_ {n}}$${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}) ^ {*}}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle n}$${\ displaystyle \ phi (n)}$

The endomorphism ring of the cyclic group is isomorphic to the ring , and the automorphism group is isomorphic to the unit group of , and this is isomorphic to the cyclic group . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ left \ {+ 1, -1 \ right \}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle C_ {2}}$

### Algebraic properties

Is a natural number, then is cyclic if and only if is equal to or for a prime number and a natural number . The generators of this cyclic group are called primitive roots modulo . ${\ displaystyle n}$${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}) ^ {*}}$${\ displaystyle n}$${\ displaystyle 1,2,4, p ^ {k}}$${\ displaystyle 2p ^ {k}}$${\ displaystyle p> 2}$${\ displaystyle k}$${\ displaystyle n}$

In particular, for every prime number the group is cyclic with elements. More generally, every finite subgroup of the multiplicative group of a field is cyclic. ${\ displaystyle p}$${\ displaystyle (\ mathbb {Z} / p \ mathbb {Z}) ^ {*}}$${\ displaystyle p-1}$

The Galois group of a finite field extension of a finite field is a finite cyclic group. Conversely, for every finite field and every finite cyclic group there is a finite field extension with a Galois group . ${\ displaystyle K}$${\ displaystyle G}$${\ displaystyle L / K}$${\ displaystyle G}$

## Remarks

1. Because the zero can not be inverted , this multiplicative connection is never (except in the trivial case of the zero ring ) a group connection for the basic set - and it cannot give it a cyclic group structure. (The prime residue class groups , which have at least one less element, are something different .)