Group action

Through a group operation, action or effect the elements of a group are identified with self-maps of a quantity in mathematics that always the product of two group elements is identified with the sequential execution of the accompanying illustrations. The amount along with the operation of on - ie amount operating group is called transformation group. The group operation makes it in algebra, geometry, and many other areas of mathematics to describe the symmetries of objects using symmetry groups. Here is the study of the set on which the operation acts, in the foreground, and the operating group is often given from the outset as a group of pictures. On the other hand, the operation of a given group provide important information about the structure of the operating group on suitably chosen quantities in group theory. It is the study of the operating group in the foreground.

3.1 railway
3.2 Transitive and sharply transitive operations
3.3 Homogeneous operations
3.4 stabilizer

5.1 Operation of a group on itself
5.2 conjugation
5.3 automorphism group of a field extension
5.4 modules and vector spaces
5.5 Categories

Introductory Example: Operation of the symmetry group of a cube to the space diagonals

ABCDEFGH be the corners of a cube in the common name, i.e., EFGH ABCD, and have opposed surfaces (see the first image ). The rotation of the die around the axis connecting the centers of these two surfaces connects (the second image ) induced the following permutation of vertices:

By the rotation of the space diagonals are swapped, namely

There are also pictures of symmetry of the cube that do not mix up the space diagonal to each other, namely the point reflection at the center ( third picture ): It corresponds

Initially, each space diagonal is indeed mirrored, but mapped to itself. They say: The group of symmetry pictures of the cube operates on the amount of space diagonals. This circumstance makes it possible to draw conclusions about the group. For this purpose, it is found that, for every pair of space diagonal symmetry figure that swaps these two and the other two can be determined, namely the reflection in the plane containing the other two body diagonals. From the general properties of the symmetric group thus follows that there is any permutation of the space diagonals of a corresponding symmetry mapping. Since there are exactly two of these permutations and symmetry illustrations that leave to set all the space diagonal (namely, the identity and the above point reflection ), one can conclude that there is a total

Symmetry are pictures of the cube to know individually without any of them. ( For a more detailed analysis of the group structure, see octahedral group. )

Definition

( Left ) Action

A (left ) operation, ( left ) action ( left ) action of a group on a set is an external binary operation

With the following properties:

For all
For all and the neutral element of

The amount is amount called and operates ( from the left) on

From the two axioms it follows that for each transformation is a bijective mapping ( the inverse map is ). That is why a group operation of on essentially the same as a homomorphism of symmetric in the group

Legal action

Analogous to the left operation is a right action, action or effect an external binary operation

With

For all
For all and the neutral element of

The difference between left and right operations lies in the manner in which such links operate on. At a left-hand operation first and then operates during the operation in a right order is reversed. From a right action is a left operation can be constructed by surgery instead of left from right. For every legal operation a left operation

Because

And

In a similar way, a left- to a right action to convert. Since not distinguish left and right action, in essence, links operations are considered from here just yet.

Terms related to group operations

Train

It is the (left- ) operation of a group on a set for each talk is

Railway, or the orbit of the tracks form a partition of the number of elements of a track ( or its width) is also called the length of the track.

The orbits are the equivalence classes with respect to the equivalence relation:

If there is one, is valid for.

The set of equivalence classes is called web space or quotient space.

For a right action is defined analogously

And

Transitive and sharply transitive operations

This is referred to as the group operation of ( simply ) transitively or says " the Group operates ( simply ) transitively on " if there are any two elements a, such that. In this case there is only a single path, which includes completely. If the group element uniquely determined with beyond by any two elements, it is called the group operation in focus ( simple) transitive.

Is there even any pair of prototypes with and each pair of images a group element for which and is then called the group operation doubly transitive and sharply doubly transitive if it is always exactly one group element with the specified property.

General determines an operation of the group for an operation always

By the ordered subsets of elements with (k- tuples)

Is ( sharp ) simply transitive, then the group operation is called ( sharp ) - fold transitive. In other words, the group operates via iff - fold transitive on when regarding only one track ( namely itself), has sharp - fold transitive if (k- tuples) of this path is always available for elements with exactly one group element. Important applications have such ( sharp ) transitive operations in geometry, see for example affinity ( mathematics), Moufangebene, Affine translational level.

Homogeneous operations

A generalization of the operation is the transitive -fold -fold homogeneous operation. A group operates fold homogeneously on the crowd with if there is always at least one group element for any two subsets, each with exactly elements, which is on maps, so with each -way transitive operation also fold homogeneous. Of the homogeneous operation is not required in contrast to the transitive operation that the predetermined original image elements are mapped in a specific order to the predetermined pixels.

Stabilizer

For a called

The stabilizer, the isotropy that Fixgruppe or subgroup of a group of operating on. By the operation of a canonical bijection between the web area ( cosets, see below) of the stabilizer and the web is then given by:

Operated ( by restricting ) on Is this surgery - fold transitive and so the operation of on even - fold transitive.

Is a subset, and a subgroup, and is

It is said that under stable or that is stabilized by. It is then always even, the stabilizer of a point is thus the maximal subgroup of the stabilized.

Follows from a that is true, it means the operation free. This is equivalent to saying that all stabilizers are trivial, ie for all

Follows from all that is true, it means the operation faithful and effective. This is equivalent to saying that the corresponding homomorphism is injective. In this case, and if in addition, the quantity ( and thus the group) is finite, it is also said, " is operating as a permutation on "

Every free group action is faithful.

If an additional amount with a links and operation is a mapping such that for all and for all:

Then is called a - equivariant or as a homomorphism of sets.

Properties

Operating the group to be the paths form a decomposition of the means: the two paths are disjoint or identical, and each element is of a sheet. Because you can define the following equivalence relation "":

The equivalence classes of this relation are precisely the orbits. It follows the

More precisely, the

From this bijection follows for a finite group, the web formula

In particular, the length of each path is a divisor of the order of

Examples

Operation of a group on itself

The simplest example of an operation is the operation of a group on itself: is always an operation on, and for

The mapping assigns to each group element to the left translation with this. is an injective group homomorphism, we obtain from this the

The same also applies for the right translation

If one considers a subset of then operates on the path of a member is then called also right coset and left coset of Note that in general need not be. The cardinality of the set of all right cosets is referred to with

Being in a group any legal translation is a bijection, for every It follows the train of the equation

One can show that there are as many left cosets as right cosets, ie that

A subset of is called a normal subgroup if and only if for all. Is a normal subgroup of then is

A shortcut on defined, with the is a group, they are called the factor group of modulo

Conjugation

A group acts on itself by conjugation of the automorphisms are called inner automorphisms, the set of all inner automorphisms is denoted by.

Automorphism group of a field extension

If a field extension, then we denote by the group of all automorphisms of the leave pointwise fixed. This group operated on by each train consists of the in -lying zeros of a polynomial with coefficients in the above is irreducible. Elements of the same orbit are called conjugated through here they have the same minimal polynomial over

Modules and vector spaces

A - (left ) module is an abelian group on which a group (from left) operates such that in addition the (left ) surgery left compatible with, ie it is

The transformations then form the group of automorphisms and the picture is a group isomorphism.

In particular, if the scalar multiplication of a vector space over the field then operates the multiplicative group