Presentation of a group
In mathematics, the presentation (or presentation ) of a group given by a set of elements that create the group, and a set of relations that exist between these producers. For example, the cyclic group of the order is generated by an element of the relation. Such a presentation are therefore also called presentation by generators and relations. Detail this means the following:
- Each member of the group can be written as a product of the specified generator ( and its inverse ).
- Two such spellings of the same element differ only in the specified relations ( and their consequences).
Each group can present itself in this way, and thus are presentations a universal tool to construct groups and investigate. Many infinite groups allow a finite presentation and thus an efficient description. The combinatorial group theory studies groups with the help of their presentations, and this provides extensive techniques available.
- 4.1 Presentation of a given group
- 5.1 truth table of a finite group
- 5.2 Cyclic groups
- 5.3 dihedral
- 5.4 Quaternionengruppen
- 5.5 Symmetric Groups
- 5.6 Coxeter groups
- 5.7 surface groups
- 7.1 The word problem
- 7.2 The Konjugationsproblem
- 7.3 The isomorphism
Motivation and History
To calculate virtually in a group, it is often helpful to rely on a cleverly chosen set of generators. This is especially true if the group is large and complicated ( or even infinite), but is produced by a small, clear amount (in the best case finally). The corresponding idea for vector spaces over a field leads to the concept of the base, which is essential for the linear algebra.
For any group one can not expect such a simple structure in general: To specify the calculation rules in the group, you have to specify the relations between the generators. These depend on the considered group and can be arbitrarily complicated. In this practical sense, the concept of presentation has been used since the early days of group theory, albeit initially without precise definition. Calculations with generators and relations can be found in the second half of the 19th century, for example in the works of Arthur Cayley, Henri Poincaré and Walther von Dyck. Only in the 20th century, the practice of finitely presented groups was expanded into a theory, combinatorial group theory, was largely initiated by Max Dehn.
Introductory Examples
The simplest case of a presentation is obtained for the group of integers with their addition. This group can be generated by a single element which is or. In this case, there are no relations, and this one writes as
Each element of writes clearly than. In the absence of any relations we also speak of the free group on the given generators.
Let us add now a the relation, where, we obtain the group
Again, you can turn any element of writing than with. However, it should also, and as a consequence for all. It follows that the group has exactly elements. It is called the cyclic group of order, and it is isomorphic to.
Universal design
If any producer and relations you claim, then initially it is not clear whether and how this group can be defined. The following design solves this problem by defining the group represented as a quotient of a free group:
Given a set whose elements we want to use in the following as a producer. It should be the free group on. This consists of all reduced words with factors, for all, and exponent, where for all. It should also be about a lot of those words. We denote the set of all elements conjugate with and. It is generated on the amount subgroup of. We call the set of all consequences of the relations.
After construction is a normal subgroup of the free group. We therefore obtained as a quotient group
And call it the group with generators and relations. Specifically called the couple's presentation, and presented by group.
Speech
In the above construction, we consider the elements of commonly referred to as elements of the group. Formally speaking, however, they are elements of the free group and not the quotient. However, it is often more convenient to consider means of Quotientenhomomorphismus as a producer of. If no confusion is to be feared, it therefore does not distinguish in between the element and its image.
Spellings
Are and finite sets, it is called the presentation at last. In this case, as presented group to write simply.
Often one writes a relation in the form in order to emphasize that this is reflected in the ratio of the neutral element. More generally, one uses the more convenient notation instead of the relation.
Universal property
Let be a set and let a lot of words about. The so- presented group has the following universal property:
In other words, the group is the " freiest potential " of generated group under the given relations. This universal mapping property is to be with the given definitions are equivalent. Each of the two characterizations can therefore be used as a definition of the group, and can be found in the literature both approaches. The respective other characterization is then an inference.
Presentation of a given group
If a group is given, we can choose a generating set of elements. The free group then allows a surjective group homomorphism with for all. Second, we can now choose a subset that generates the core as a normal subgroup. Thus we get a group isomorphism. This presents the group given by the producers and the relations existing between them. Note here the trick that the relations are expressed in terms of the free generators, which are used here as variables or placeholders for the actual group elements.
If one can choose a finite generating system, called finitely generated. If you also can choose a finite set of relations, then called finitely presented.
Examples
Truth table of a finite group
Is a finite group of order, so we can interpret their truth table as a presentation by generators and relations. The producers here are the elements of the given group, and each product defines a relation in the free group on. In general, however, allows much shorter presentations, as the following examples show.
Cyclic groups
The presentations and have already been presented above as introductory examples. Each presentation with only one producer defines a group isomorphic to this.
Presentations with two generators can be surprisingly complicated, however, already. Two particularly simple examples are given by the dihedral and the quaternion group.
Dihedral
The dihedral group of order is the isometry group of a regular -gon in the plane. It is generated by two adjacent reflections, we obtain the presentation
Quaternionengruppen
The generalized quaternion group of order for is given by the presentation
For we obtain from this the Hamiltonian quaternion with the link
In this case the spelling and and and is historically common.
Symmetric groups
The symmetric group is generated by transposition, in which. It is calculated by directly that the following relations hold between these generators:
- For all
- If
- If
The group presented as
Thus allows a surjective group homomorphism virtue. It is not immediately apparent that this is also injective, that the relations specified already generate all relations. However, it can be shown using the above relations, that contains not more than elements, and thus is considered.
Note that one can rewrite because of the above relations as
- For,
- For.
Also, this equivalent notation is commonly found in the literature.
Coxeter groups
Reflection groups are such groups that are generated by reflections, that is, elements of atomic. Reflection groups play an important role in classical geometry, for example, in the classification of regular polyhedra. They have been extensively studied by the British mathematician Harold Scott MacDonald Coxeter, in whose honor they are also called Coxeter groups.
In order to write down all the relations of such a group clearly, we choose a symmetric matrix whose entries are natural numbers or infinite, ie for. We take this to additionally that and for all. Such a matrix is then called Coxeter matrix and defines the following Coxeter group:
If so, the corresponding relation is simply omitted.
For example, the dihedral group is the Coxeter group to the matrix
The symmetric group is the Coxeter group to the matrix
Such matrices can be clearly represented as Dynkin diagrams and classify.
Surface groups
The fundamental group of the closed orientable surface of genus has the presentation
Tietze transformations
There are always infinitely many different presentations of a given group. For example, change the following transformations, the presentation but not presented group:
The set of Tietze states that these transformations already exhaust all possibilities:
The three Dehnschen problems
The German mathematician Max Dehn has marked the beginning of the 20th century with its fundamental work the combinatorial group theory crucial. It has been found here in particular three general problems for working with presentations of fundamental importance, both in practical and theoretical ways.
The word problem
The first problem is the most obvious: If you want to calculate concrete in the group, then you have to compare elements and to determine whether they are identical or different. Since all elements can be written as words on the generating sets, you will immediately on the following word problem:
For this purpose, the following problem is equivalent, by:
So after constructing one must determine whether or not located in the normal subgroup. Even with a small set of relations, however, the normal subgroup generated in this way is huge. At least you can count the amount systematically and thus the word problem is always semi- decidable: If applies, then you will find this after last a long time as a consequence of the relations. Applies the other hand, then finds the list of no end.
The set of Novikov - Boone says that the word problem in general is algorithmically unsolvable.
The Konjugationsproblem
The Konjugationsproblem similar to the word problem, but it is even more difficult in general:
With this one contains the word problem as a special case.
Just as the word problem, the Konjugationsproblem is only semi- decidable and in general algorithmically unsolvable.
The isomorphism
The third and most difficult of the problems Dehnschen is the isomorphism:
Describe the above explained Tietze transformations, how to transform presentations into one another. Starting from a given presentation can thus enumerate all equivalent presentations. Just as the word and Konjugationsproblem the isomorphism problem is only semi- decidable and in general algorithmically unsolvable.