In mathematics , a group operation, action or effect includes a group as the “active” part and a set as the “passive” part. The operation, action or effect of an element on the set is a transformation ( self-mapping ) of this set. Thereby operate the elements to the elements of the set in such a way that the action of the product of the sequential execution corresponding to the individual actions.
The operating group is called the transformation group . The quantity together with the operation of on is called -quantity.
If additional structure is important to the set , be it algebraic , geometric , topological , a group operation will only be considered admissible if it preserves this structure.
The group operation makes it possible in algebra, geometry and many other areas of mathematics to describe the symmetries of objects with the help of symmetry groups . The focus here is on examining the crowd on which the operation is acting. On the other hand, the operation can be a predetermined group to suitably chosen amounts in the group theory provide on the structure of the operating group important information. In this case, the focus is on examining the operating group.
Introductory example: cube group and room diagonals
let the corners of a cube be in the usual notation, i.e. i.e., and are opposite surfaces (see first picture). The rotation of the cube around the axis that connects the centers of these two surfaces (second picture) induces the following reversal of the corners:
and at the same time
The rotation also (at the same time) swaps the 4 room diagonals, namely
Another image of symmetry, the reflection on the plane (fourth picture), leaves the 2 room diagonals and fixed and swaps the other 2
But there are also symmetrical images of the cube that do not interchange the spatial diagonals, namely the point reflection at the center (third picture): it corresponds
and at the same time
and at the same time
and at the same time
Every single room diagonal is mapped onto itself, even if it is mirrored.
One says: The group of symmetry images of the cube (called the “ cube group ”) operates on the set of corners, on the set of edges, on the set of space diagonals, etc. In order to capture this group, the focus will be on the permutations in the following the room diagonals directed.
For each pair of space diagonals there is now a level reflection (in the figure for pair and ), which swaps these two and leaves all other space diagonals fixed, namely the reflection on the level that contains the fixed space diagonals. Such a pair exchange is called transposition , and these transpositions produce the whole symmetrical group of permutations of the (four) space diagonals. Since there are these permutations and exactly two symmetry mappings that fix all space diagonals (namely the identity and the point reflection mentioned above), one can conclude that there is a total of
There are symmetry images of the cube without knowing each of them individually. (For a more detailed analysis of the group structure see octahedral group .)
A (left) operation, (left) action or (left) effect of a group on a set is an outer two-digit link
with the following properties:
for everyone , where the neutral element of is ("identity"),
for everyone ("compatibility").
One then says operate on (from the left) and together with this group operation name a (left) set.
From the two requirements it follows that for each the transformation is a bijective mapping (the inverse mapping is ). Therefore, the action of a group element is not just a self-mapping, but a permutation of , and a group operation from on can be equated with a group homomorphism from in to the symmetric group .
Analogous to the left operation, a right operation, action or effect is an outer two-digit link
for everyone and the neutral element of
The difference between left and right operations lies in the way as shortcuts to operate. A left operation operates first and then , while a right operation reverses the order.
A left operation can be constructed from a right operation by writing the operation as a left operation of the opposing group , or by operating from right instead of left . There is a left operation for every right operation
A left operation can be converted into a right operation in a similar manner. Since left and right operations do not essentially differ, only left operations are considered from here on.
Let it be the (left) operation of a group on a set For each one then calls
the web, the transitivity, the Transitivitätssystem or the orbit (engl. orbit ) of the tracks form a partition of the number of elements of a web (or their cardinality ) is the length of the web mentioned. For a fixed one you call the through
given image the "orbit image".
The orbits are the equivalence classes with regard to the equivalence relation:
if there is one for which applies.
The set of equivalence classes is called orbit space or orbit space .
For a right operation one defines analogously
Be a topological space and a transformation group of . For a point denote the orbit of . Then the set is called a fundamental domain of if the cut for each is a one-element set.
The square is a fundamental domain of with respect to the transformation group . Each point can be written with and .
Transitive and sharply transitive operations
It refers to the group operation of on the (simple) transitive or says "the Group operates (simple) transitively on " when it comes to two elements one is so true. In this case there is only a single path that encompasses the whole . Is the group element with addition by any two elements clearly determined, it is called the group operation sharp (simple) transitive.
If there is even for every pair of archetypes with and every pair of images with a group element for which and is, then the group operation is called double transitive and sharply double transitive, if there is always exactly one group element with the mentioned property.
If misunderstandings are not to be feared, instead of the formulation “the symmetry group of the graph operates transitively on the edges” the shorter “the graph is edge-transitive ” or “the group is edge-transitive” can be used .
General surgery determines the group on for ever surgery
on the ordered subsets of with elements ( k -tuples with different components in pairs)
If (sharp) is simply transitive, then the group operation is called (sharp) -fold transitive. In other words, the group operates via if and fold transitive on when regarding only a web (namely , sharply itself) has fold transitive if it elements ( k tuple) this path always exactly one group element with there. Such (sharp) transitive operations have important applications in geometry , see for example affinity (mathematics) , Moufang level , affine translation level .
- The group of four operates (sharply simply) transitively on the set , since the number 1 can be converted into any other. This does not apply to the group of four , which is isomorphic to .
- The Galois group of an irreducible polynomial with rational coefficients operates transitively on the set of zeros of the polynomial.
Intransitive permutation group
If the group operation has more than one orbit, it is called intransitive. Those permutations of an intransitive permutation group that only swap the digits of a path, leaving the other digits unchanged, form a subgroup that becomes transitive if the unchanged digits are left out.
A generalization of the -fold transitive operation is the -fold homogeneous operation. A group operates fold homogeneous on the set with if for any two subsets , each with exactly always at least one group element elements are that on , so with maps Each fold transitive operation is also homogeneous fold. In contrast to the transitive operation, the homogeneous operation does not require that the specified original image elements be mapped onto the specified image elements in a specific order.
For one is called
the stabilizer, the isotropy, the Fixgruppe or subgroup of a group of which to operate. The operation then gives a canonical bijection between the orbit space (secondary classes, see below) of the stabilizer and the orbit of :
operates (by restricting ) on If this operation is -fold transitive and so the operation on is even -fold transitive.
Is a subset and a subgroup, and holds
it is said that stable under or that of stabilized is. It is then always true that the stabilizer of a point is the maximum subgroup of the stabilized.
Free and faithful operations
The operation is called free if each element of the set is only fixed by the neutral element of the group. This means that all stabilizers are trivial; H. for all
The operation is called true or effective if only the neutral element of the group fixes all elements of the set. This means that the associated homomorphism has a trivial core , i.e. is injective . For faithful operations can be viewed as a subgroup of . For faithful operations with finite sets one also says: " operates as a permutation group on "
Any free group operation on a non-empty set is faithful.
Homomorphisms between G sets
If there is another set with a -link operation and a mapping such that for all and for all :
then it is referred to as G equivariant or also as homomorphism of sets .
The equivalence classes of the equivalence relation introduced above are exactly the orbits. From this follows the
Orbital equation: The thickness of is equal to the sum over the length of all orbits.
More precisely applies (with as the fixed group of ) the
Path set : Ifthe imageisthena bijection .
From this bijection the orbit formula
follows for a finite group
In particular, the length of each trajectory is a factor of the order of
Operation of a group on itself
Operation by multiplication
The simplest example of an operation is the operation of a group on itself: is always an operation on , for and
The mapping assigns the left translation to each group element with it. Because the operation is faithful, an injective group homomorphism is obtained from this
Cayley's theorem : Every finite group of the orderis isomorphic to a subgroup of the symmetric group
The same applies to legal translation
If one considers a subgroup of then also operates on the path of an element is then also called right secondary class and left secondary class of. Note that generally does not have to be. The power of the set of all right secondary classes is called
Since every right translation in a group is a bijection, it follows for every one of them with the equation of the orbits
Euler-Lagrange theorem : For every subgroup ofa finite groupwe have:
- In particular, the order of is a factor of the order of
One can show that there are just as many left secondary classes as there are right secondary classes, so that
A subgroup of is called normal subgroup if applies to all . Is a normal divisor of then becomes through
a link on defines with which a group is, they are called the factor group of modulo
Operation by conjugation
A group operates on itself through conjugation , so
In this context , the orbits are referred to as conjugation classes , the stabilizers as centralizers . In this case the class equation is obtained from the orbit formula .
The automorphisms are called inner automorphisms , the set of all inner automorphisms is denoted by.
Automorphism group of a body extension
If there is a body extension , then one designates with the group of all automorphisms of which leave pointwise fixed. This group operates on through. Each path consists of the zeros lying in a polynomial with coefficients in which over is irreducible. Elements of the same orbit are called here conjugated over they have the same minimal polynomial over
Modules and vector spaces
A - (left) module is an Abelian group on which a group (from the left) operates, so that the (left) operation is also left compatible with , i.e. i.e., it applies
for everyone and everyone
The transformations to then form the group of the automorphisms on and the picture is a group isomorphism .
If in particular the scalar multiplication of a vector space over the body then the multiplicative group operates on
Is general an object of any category , so may a structure acceptable operation of an (abstract) group to be defined as a group homomorphism
this is the group of automorphisms of the category theory sense. The operations of groups on sets or Abelian groups mentioned above are special cases.
↑ Fundamental area . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
^ Nieper-Wißkirchen: Galois theory. , University of Augsburg (2013), consequence 4.11, p. 133 ( online ).