Cyclic group
In group theory, a cyclic group is a group which is generated by a single element. It consists of all the powers of the generator:
A group is cyclic if it contains an element ( the "producer " of the group ), so that each element of a power of being. Synonymous with this is that there is an element, so that itself is the only subgroup of that contains.
Cyclic groups are the simplest groups and can be completely classified: For every natural number there is a cyclic group with exactly elements, and there is the infinite cyclic group, the additive group of integers. Any other cyclic group is isomorphic to one of these groups.
- 3.1 Subgroups and Factor Groups
- 3.2 Endomorphisms and automorphisms
- 3.3 Algebraic properties
Illustration
Rotation groups
The finite cyclic groups can be illustrated as rotation groups of regular polygons in the plane. For example, the group consists of the possible rotations of the plane, the convict a given square in itself.
The figure above shows a square A and the positions B, C and D, in which it can be transferred by rotating. Among the necessary rotation to be specified in each case. The elements of the cyclic group here are the movements and the positions of the square. That is, the group is in this illustration of the set { 0 °, 90 °, 180 °, 270 ° }. The combination of the elements is the sequential execution of rotations; this corresponds to an addition of the angle. The rotation through 360 ° with the rotation of 0 ° corresponds to angle Strictly speaking modulo 360 ° added.
Allows you to not only rotations of the plane, but also reflections, then you get the so -called dihedral groups in the case of polygons.
Note also that the rotating group of the circle S1 is not cyclic.
Residue class groups
A different representation of a cyclic group provides the addition modulo a number, the so-called residual arithmetic. In the additive group is the residue class of 1, a generator, which means you can get any other residue class by the 1 repeatedly added to itself. In the example, this means that all four elements can be represented as the sum of 1, ie, 1 = 1, 2 = 1 1, 3 = 1 1 1, 0 = 1 1 1 1. The residue class group behaves the same as the rotation group described above {0 °, 90 °, 180 °, 270 ° }: 0 corresponds to 0 °, 90 ° 1 corresponds etc: These two groups are isomorphic.
Spellings
For finite cyclic groups, there are essentially three formats: , and. For non- finite cyclic groups, the two spellings and available. As a group operation is usually in the multiplication and in, and uses the addition.
The spellings, and stem from the fact that the additive group of residue class rings and are the best known representatives of cyclic groups by themselves. However, unlike the notation lead them to accept the presence of a ring structure. The term is also still used in conjunction with p-adic number.
Properties
All cyclic groups are abelian groups.
A cyclic group can have multiple producers. The producers of 1 and -1, the producers of the residual classes that are relatively prime; their number is specified by the Euler φ function.
Is generally d is a divisor of n, then the number of elements that are of order d:
.
The direct product of two cyclic groups is cyclic if and only if n and m are relatively prime; in this case the product is isomorphic to.
Every finitely generated abelian group is a direct product of finitely many cyclic 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 residue class ring isomorphism is the discrete logarithm: Is a generator of, then the map
An isomorphism.
Subgroups and factor groups
All subgroups and factor groups of cyclic groups are cyclic. In particular, the sub-groups of the form with a natural number m are cyclically. All of these sub-groups are different, and for m ≠ 0 are isomorphic to.
The association of subsets of is isomorphic to the dual association of natural numbers with the divisibility. All factor groups are finite, except for the trivial factor group.
For each positive divisor d of n, the group has exactly one subgroup of order d, namely the subgroup generated by the element d n / { kn / d | k = 0, ..., d- 1}. Other than these sub-groups do not exist. The sub-group association is therefore isomorphic to the divisor Association of n
A cyclic group is simple if and only if its order is prime.
Endomorphisms and automorphisms
The endomorphism ring (see group homomorphism ) of the group ring is isomorphic to the residue class ring. Under this isomorphism corresponds to the residue class r of the endomorphism of which maps each element to its r -th power. It follows that the automorphism group of is isomorphic to the group of the unit group of the ring is. This group consists of the elements that are relatively prime to n, and has φ ( n ) elements.
The endomorphism ring of the cyclic group is isomorphic to the ring, and the automorphism group is isomorphic to the unit group { 1, -1 } of, and this is isomorphic to the cyclic group C2.
Algebraic properties
If n is a natural number, then exactly then is cyclic if n is 2, 4, or is, for a prime p > 2 and a natural number k, the generator of this cyclic group are called primitive roots modulo n
In particular, for every prime p, the cyclic group with p- 1 elements. Generally, any finite subgroup of the multiplicative group of a body is cyclic.
The Galois group of a finite field extension of a finite field is a finite cyclic group. Conversely, there is for each finite field K and any finite cyclic group G be a finite field extension L / K with Galois group G.