Permutation group

In group theory is called a group of permutations of a finite set M with the sequential execution as a group linking permutation. The group of all permutations of M called its symmetric group. The permutation groups in this sense are exactly the subgroups of the symmetric groups.

By the theorem of Cayley every finite group is isomorphic to a subgroup of the symmetric group to a permutation. In this respect "is" every finite group is a permutation group. Looking at the finite group G as an abstract algebraic structure, then it is said, therefore, more precisely, G acts as a permutation on the set M. It is thus clear that there is a clear description of the group structure in this faithful permutation representation, in addition to the other descriptions are possible.

• 3.1 Finite symmetry groups
• 3.2 automorphism groups of finite structures

Definitions

Defined by a group operation

Let be a group with the neutral element e G acts as a permutation on M if and only if:

A group operation that meets only the second and third condition is, true. So G acts as a permutation on M if and only if the operation is faithful and M finite. A group operation, which only satisfies the first and second condition is, as permutation (English: permutation representation ) designated by G. So G acts as a permutation on M if and only if the group operation is a faithful permutation representation.

Defined by a group homomorphism

Equivalent Description: G acts as a permutation on M if and only if M is a finite set and an injective group homomorphism exists. It is therefore the set of all bijective self-maps of the set M. In this description, the operation of the first definition is given by the requirement of injectivity is equivalent to the requirement that the operation was faithful.

Note that need not be required separately in the mentioned definitions for a permutation group, that the group G is finite; this follows from the finiteness of M.

Isomorphic as permutation groups

For two groups G and H, which operate on two finite sets M and N as permutation groups, exacerbating the Isomorphiebegriffs is defined: G and H are called isomorphic as permutation groups if and only if a group isomorphism and a bijection exists, such that for all. One can show that two groups G and H, which operate faithfully on the same amount, if and are isomorphic as permutation groups, if their particular by the group operations image groups are conjugate subgroups in the symmetric group, that is, if they by conjugation fixed with a group element can be mapped to each other.

Semi Regular and regularly

• If G acts on M as a permutation group, this operation is exactly then called semi- regular and semiregular permutation group G if the only element of G that fixes any element of M, the identity element of G is. Formal:

• The operation is said to be regular and called G if a regular permutation group on M, if the operation is semi- regular and transitive. The operation is transitive, when each element of M can be represented by the operation on any other element of M. Formal: See about other possible Transitivitätseigenschaften a permutation group operation # Transitive group operation.

In the term (left ) regular representation and also in the specialized groups sense of the word, as it is described in the article Cayley, describes homonym regularly as a property that neither specialized nor generalizes the one described here! The invention described in the first sentence of Caily " special regular representation ", in which the group operates via Linksmutiplikation on itself actually - perhaps by accident - a but in general not the only "regular permutation representation " of the group. This special case is illustrated in the examples in this article.

Properties

The properties described in this section can be found in the textbook Design Theory, which is called in the literature. Trivial properties are demonstrated here or in the Examples section and counter-examples in this article.

• Every finite group allows a representation as a regular permutation group. Such a representation is given by the " left multiplication " of the group on itself, see the examples.
• For every finite group can be explained as the group operation on any finite set M is a permutation, we choose as the trivial operation. However, a faithful permutation representation requires a dependent of the group structure minimum number m of elements. Then for any natural number n, which is not less than m, a faithful permutation representation on any set with n elements.

• Is only for the trivial one group.
• If the group G is an element of order, which is a prime power, then.
• Especially true then by the theorem of Cauchy, a special case of the Sylowsätze: Tells the prime p, the group order, then.

• Let G be a group, a subgroup. If G acts on M as a permutation group, then H also operates on the limited to this subgroup as a permutation operation on M.

• If the operation of H is transitive, then it is also of G, conversely, can be transitive, the action of G, restricted by H but not.
• However, if the action of G semi- regular, then it is also that of H, also the converse does not apply.

Examples and counter-examples

The ideas for the examples given in this section can be found according to the sense in the textbook Design Theory, which is called in the literature.

• Every finite group acts on itself by left multiplication. This operation is faithful and semi- regular ( because of the cancellation law for groups), and transitive, ie operates every finite group via left multiplication as a regular permutation group on the set of its elements and is thus isomorphic if G contains a transitive subgroup of the symmetric group n elements. The right multiplication generally leads to a different embedding of the group, must also ensure the group join are reversed: so that the right multiplication satisfies the above rules ( 2 ) for an operation from the left or the rules need for an operation of the right mutatis mutandis be reformulated.

• The cyclic residue class group operates regularly through the links Addition on himself and in the same way on the remains.

• The symmetric group on n elements operating in their initial presentation on faithful and transitive, but only for semi- regular. But to herself she operated with the left multiplication as a regular permutation group.
• A finite group G acts on itself by conjugation. This operation, however, is not true in general. Every finite, noncommutative simple group operates, however, via conjugation as a permutation (ie faithful ) to itself
• The linear group ( prime power ) operates as a permutation group. M is a finite set of vectors in the n-dimensional vector space over the finite field with q elements. The operation is transitive, but not generally semi- regular.

• Is a real linear subspace of and the subgroup that maps M as a whole to himself, then H acts transitively, but not as a permutation group on, because the operation is not true. On the other hand operates the factor group, the subgroup of G and U, which fixes each element of N, in a natural way as a transitive permutation group on.
• Although for an infinite field K (for example) operates faithful and transitive, but not as a permutation group on, because N is not finite.

• Be the Klein four-group as a subgroup of the symmetric group. operates on a regular permutation group.

• The group contains 3 more to isomorphic subgroups, eg. Since, as defined herein, semi regularly operates on M, U, however, does not and because the web 1 only contains two elements during the operation of the U, the two sub-groups not as permutation isomorphic to M. In contrast, the U with the other two (of different! ) Groups, which are generated by two disjoint transpositions, as a permutation isomorphic.
• The sub-group is as transitive, but G is not semi- regularly in contrast to H.

• The cyclic group with 6 elements operates as a regular permutation group via left multiplication on itself, which corresponds to their usual permutation on. But also operates as a permutation on the set, but not here and not transitive semi regular. The number is the minimum cardinality of a set, operating on the G as a permutation group for this group. The limited operation of is semi- regularly, but not transitive.
• The cyclic group with three elements operate on regular, their permutation may be limiting the operation of the symmetric group whose subgroup is H, are considered. But G acts transitively on M though, but not semi- regular.

Finite symmetry groups

In geometry, many groups occur which are defined by the fact that they represent a geometric figure as a whole to be. For example, the group of motions of the three-dimensional space perception, which as a whole represent the unit cube ( spanned by the three standard basis vectors ) to be a typical symmetry group.

• The symmetry group of a ( non-degenerate ) polyhedron in the space of intuition operates as a permutation group on the ( endlichen! ) set of vertices of the polyhedron.
• The symmetry group G of a ball in the space of intuition acts transitively on the set M of points on the sphere surface, but on any amount as permutation: Because the operation on M is transitive, it can not be for the whole symmetry group G on a finite set of points N limit. In contrast, the symmetry group of the unit cube can be regarded as a subgroup of G, if one chooses a ball to the cube circumscribed ball, so the ball through all the vertices of the cube.
• The symmetry group of an equilateral triangle in the real plane operates as a transitive permutation group, but not semi- regularly on the set of vertices of the triangle.
• General surgery the symmetry group G of a regular n -gon in the plane as transitive, not semiregular permutation group on the set of vertices of the n- gon. This description can be used for the definition of the dihedral group ( as a subgroup of the symmetric group ).
• The symmetry group of a distance on the real line ( ie, a real interval ) operates as a regular permutation group on its boundary points. She is the two-element group, where s is the reflection of the line at the middle of the interval.
• In contrast, the symmetry group H operates ( in the sense described above ) a path in three-dimensional space is not true and therefore not. Than the permutation group on the boundary points of the route This group is even infinitely - note the rotations, where the route lies on the axis! As in the example of a linear subspace in a finite vector space you have above for the factor group by the subgroup U of movements that represent each point on the stretch to pass. This takes you back to a group that is isomorphic to the group mentioned in the previous example. Often these canonical factor group is then used as the symmetry group ( here: the path) respectively.

Automorphism groups of finite structures

The structure-preserving, bijective self-images finite structures, for example the finite incidence structures such as block diagrams, of the finite projective planes, etc. operate as a permutation on the finite set S of the " elements" of the structure ( for incidence structures, so the amount of "points" along with the amount of "blocks" ). In important cases, such as for all simple block diagrams ( ie also for all "classical" finite geometries ), it is sufficient to use as a lot of the point or the block size, since the automorphism been operated on at least one of these quantities faithful. Mostly, the point set is used. The set of all structure-preserving, bijective self-maps of the structure is referred to as full automorphism group of the structure, each of its sub-groups as automorphism group. By construction, these groups operate as a permutation on the set of structural elements, in the most important cases mentioned already on the point set.

• The finite group operates as a simple permutation and full transitive Automorhismengruppe but not regularly on the Fano projective plane, ie specifically to the amount of their 7 points. In the article Fano plane, the structure of this group and the faithful permutation representation described here is presented in detail as a subgroup of the alternating group.
• The five sporadic Mathieugruppen operate as permutation and full Automorhismengruppen on each one assigned to them Wittschen block plan - here the set of points for the unique description is sufficient.
• A somewhat contrived example of an incidence structure in which the full automorphism group operates neither on the point still on the block quantity alone as a permutation group, is with the amounts. Here, the automorphism group is the product, so G is isomorphic to the Klein four-group. But G acts neither on the point still on the block amount faithful! The same statements apply if you defined for this point and block the amount of incidence rather than by.