Alternating group

The alternating group of degree consists of all even permutations of a - element set. The linking of the group is the concatenation ( sequential execution ) of permutations. Most will simply speak of the alternating group.

The alternating groups are subgroups of the corresponding symmetric groups. Particular importance is attached to the alternating group. That she is the only non -trivial normal subgroup of, is an important part of the proof of the theorem of Abel- Ruffini. This sentence from the beginning of the 19th century states that the fifth or higher degree polynomial equations can not be solved by root expressions.

• 4.1 transpositions
• 4.2 transpositions and inversion number
• 4.3 transpositions and seclusion

Properties

The alternating groups are defined only for.

The alternating group consists of (half factorial) elements. Only the groups are abelian. The alternating group is the commutator subgroup of the symmetric group.

Up to and all alternating groups are simple. is the smallest non-abelian simple group; it is isomorphic to the rotation group of the icosahedron (see icosahedral group ).

Generating system

The alternating group is generated by the 3 -cycles of the symmetric group.

Each 3-cycle is an even permutation, since it is the product of two transpositions

Write leaves, and therefore an element of the alternating group. Furthermore, every even permutation is a product of 3 -cycles, as a pair of two transpositions are products of 3 -cycles. Is considered in detail

• When both transpositions are equal.
• When both transpositions have a common element.
• When both transpositions have no common element.

Inversions and inversion number, even and odd permutations

A failure status or an inversion occurs when two " points" of a permutation in "wrong" order stand. To determine the inversion number of a permutation of all its points are pairwise compared and the number of inversions is counted.

For example, the permutation in Tupelschreibweise having the inversions, " 0:57 " and " 1:57 " (indicated by two-line form ) and thus the number of inversion.

From a straight permutation is when the inversion number is an even number, by an odd permutation is when the inversion number is an odd number.

Often one also defines the sign as follows:

The Signum is a group homomorphism, it is thus:

For the permutations and.

Group Properties

As the core of the sign is automatically a normal subgroup of. One can easily recalculate the subgroup characteristics:

For the set of even permutations applies:

• The identity permutation is an element of this set.
• The amount is closed under concatenation, ie if and straight permutations are even and especially, a proof sketch is given below.

With these conditions, "inherits" directly from all of the necessary group properties:

• For all even permutations applies:
• For all even permutations applies:
• For all even permutations is true: there is a straight with

The group in this case represents a special feature, because it is the smallest, simple, non - abelian group.

Seclusion

Transpositions

As transposition is called a permutation, in which exactly two different digits are interchanged, eg Be reversed at 3 and 5.

In general, for all n- digit permutations and: can be generated with a finite number of transpositions.

As a special case of this is true for an arbitrary permutations can be generated with a finite number of transpositions of the identical permutation.

When choosing the necessary transpositions some freedom exists, one could right in the picture as Trans positions b and c can be omitted, since they cancel obvious. Likewise, you could use the number of transpositions to 7, 9, 11, ... increase by the incorporation of additional pairwise aufhebender transpositions. However, it is not possible to generate an even number of transpositions from.

Transpositions and inversion number

By a single transposition is the value of the inversion number changes is always an odd number, i.e., in a straight permutation is odd, and vice versa.

In a transposition that from   the new permutation   generated, the change of the inversion number is composed of the sum of the following amendments:

• Change that results from the new sequence of x and z, it is 1, if x
• Change that results from the new sequence of x, y and z. if yi largest or smallest element of x, y, z, is the change 0
• If yi central element of x, y, z, the change of 2 or -2.

The sum of an odd and as many even numbers always gives an odd number.

The further statement made above can be generalized:

• By an odd number of transpositions, the value of inversion number changes is always an odd number, that of a straight permutation is an odd, and vice versa.
• With an even number of transpositions, the value of the inversion number is always changing to an even number, ie of an even permutation is again an even permutation and an odd permutation is again an odd permutation.

Transpositions and seclusion

Since id is an even permutation, the following applies:

• All even permutations can produce precisely the number of transpositions from id only by a.
• All odd permutations can be generated only by an odd number of transpositions from id.

If p and q are even permutations, then there are even numbers and so that p and q as the concatenation of transpositions can be represented as follows:

This applies, thus the concatenation is straight.

Analogously, we derive: the concatenation of an even and an odd permutation always produces an odd permutation. This results in the assumption that a permutation is even and is odd due to the contradiction.