Group (mathematics)

In mathematics, a group is a set of elements together with a link to a third element assigns the same amount each two elements of the set, while three conditions, the group axioms are satisfied. These include the associative law, the existence of a neutral element, and the existence of the inverse elements. One of the best known groups is the set of integers with addition as operation. The mathematical subspecialty that the study of the structure group is dedicated to is called group theory. It is a branch of algebra. The areas of application of the groups, even outside mathematics, making it a central concept of current mathematics.

Groups share a fundamental kinship with the idea of ​​symmetry. For example, embodies the symmetry group of a geometric object whose symmetrical properties. It consists of the set of those images (eg rotation), which leave the object unchanged, and the consecutive execution of such pictures as a link. Lie groups are the symmetry groups of the Standard Model of particle physics, point groups are used to understand the chemical symmetry at the molecular level; and Poincaré groups can express the symmetries underlying the special theory of relativity is based.

The concept of group originated from Évariste Galois's investigations of polynomial equations in the 1830s. After contributions from other fields of mathematics such as number theory and geometry of the term group has been generalized. By 1870 he was firmly established and is now treated in the independent field of group theory. To explore groups, mathematicians have developed special terms to decompose groups into smaller, more understandable components, such as subgroups, factor groups and simple groups. In addition to their abstract properties group theorists also examine ways in which groups can be expressed in concrete terms ( representation theory ), both for theoretical investigations as well as for concrete calculations. A particularly rich theory has been developed for finite groups, culminating in the classification of finite simple groups in 1983. These play a similar role as the prime natural numbers for groups.

• 3.1 sets of numbers
• 3.2 The trivial group
• 3.3 Cyclic groups

Introductory Example

One of the groups is the best known set of integers, which is commonly referred to.

The set of integers together with the addition meets some basic characteristics:

• For two integers, and the sum is an integer again. If one, however, divide two integers together, the result would be mostly a rational number and not an integer more. Since this can not happen in the addition, they say that the integers are closed under addition.
• For all integers, and the associative law holds

• Is again an integer number, then:

• For each integer there exists an integer, so that is true. That is for each integer exists an integer such that their sum is zero. The number is, in this case the inverse element of and is quoted at.

These four properties of the set of integers together with their addition to the definition of the group generalized to other quantities, on which there is a matching operation.

Definition

Group

A group is a pair consisting of a set and a two-digit shortcut. That is, by the image will be described. Meets the following link axioms, then group is called:

• Associativity: For all group elements, and the following applies:
• There is a neutral element, applies to the group for all elements.
• For each group element exists with an inverse element.

Equivalent characterization

The group axioms can be weakened formally by replacing the axioms for the neutral and the inverse element as follows:

There is a left identity, the following applies that:

• For all group elements applies:
• Each exists with a left inverse element.

This formally weaker definition is equivalent to the original definition.

Abelian group

A group is called abelian or commutative if in addition the following axiom is satisfied:

• Commutativity: and applies to all group members.

Otherwise, that is, when there are group elements is for those who say the group of non- Abelian ( or non- commutative ).

Order

In a group, the thickness is also referred to as the order of the group. For a finite set of the order is therefore simply the number of group elements.

Notes on notation

Frequently for the shortcut icon is used, then one speaks of a multiplicatively written group. The neutral element is then called unit element and is symbolized by. As is usual also in the ordinary multiplication, the Malpunkt can be omitted in many situations. Then also the product of characters is used for links of several elements. For the times linking a group element is written as a power to itself and to define and.

The group properties can also record additively by the icon used for the shortcut. The neutral element is then called zero element and is symbolized by. The group element to the inverse element is symbolized in an additively written group not by, but by. A times the sum will be referred to here, and are employed as well. An abelian group can be understood in this way as a module over the ring of integers. Usually, the additive notation is only for abelian groups, while not abelian, or any groups are usually written multiplicatively.

If the logic clear from the context, we write often only for the group.

Examples

Below are some examples of groups. Thus, groups of numbers, a group with exactly one element and examples are given of cyclic groups. Further examples of groups can be found in the list of small groups.

Sets of numbers

• The set of integers together with the addition forms an ( abelian ) group. Together with the multiplication, the amount of the integers, however, no group.
• The set of rational numbers or the set of real numbers is a group with the addition. Together with the multiplication, the quantities and also groups.

The trivial group

The amount of only one element may be regarded as a group. Since each group has a neutral element, exactly one element of this must then be taken as the neutral element. So then applies. Using this equality, the remaining group axioms can be proved. The group with exactly one element is called the trivial group.

Cyclic groups

A cyclic group is a group the elements of which can be represented as a power of one of its elements. Using the multiplicative notation are the elements of a cyclic group

Being said and referred to the neutral element of the group. The element is called a generator or a primitive root of the group. In an additive notation element is a primitive element, when the elements of the group by

Can be displayed.

For example, the considered in the first section additive group of integers is a cyclic group with the primitive root. This group has infinitely many elements. In contrast, the multiplicative group of the n- th complex roots of unity has finitely many elements. This group consists of all complex numbers, the equation

. meet The group elements can be visualized as vertices of a regular n- gon. For this is done in the diagram on the right. The group operation is multiplication of the complex numbers. In the right image corresponds to the multiplication by the rotation of the polygon in the counterclockwise direction.

Cyclic groups have determined to be the property by the number of its elements clearly. That is, two cyclic groups of elements are isomorphic, so it can be a group isomorphism between these two groups to be found. In particular, therefore are all cyclic groups with infinitely many elements equivalent to the cyclic group of integers.

Basic properties of a group

• The neutral element of a group is uniquely determined. Are namely, neutral elements, then must be, since it is neutral, and, as is neutral. Thus follows.
• It is the reduction rule: From or with the group elements follows each. This can be seen by.

• The equation is always uniquely solvable and the solution is. Similarly, the unique solution.
• The inverse of a group element element is uniquely determined. If and are both inversely follows then:

• It is and.
• For all elements. This follows from the equation chain

Group homomorphism

Group homomorphisms are mappings that preserve the group structure. A picture

Between two groups and is called homomorphism of groups or short- homomorphism, if the equation

Applies to all elements. Is the picture in addition bijective, it is called group isomorphism. In this case, is called the groups to each other and isomorphic.

Products of groups

In group theory, different products of groups are considered:

• The direct product is given by the Cartesian product of the carrier amounts along with the component-wise join.

• The semi- product is a direct generalization of the direct product, wherein the one of the second group operates on. It can also be implemented as an internal semidirect product of a normal subgroup and a subgroup of a given group.

• The wreath product is a special semi- direct product.

• The complex product of two subgroups of a given group is given by pairwise products of subgroup elements. This product is more general sense for any two subsets of the group.

• The free product represents the categorical coproduct in the category of groups

• The amalgamated product is a generalization of the free product, in which the elements of a common sub-group are fused together ( " amalgamated " ) are.