Alternating group

The alternating group of degree ${\displaystyle n}$ consists of all even permutations of a -elementary set. The group is linked by the concatenation (execution) of the permutations . Usually the alternating group is simply spoken of. ${\displaystyle n}$ ${\displaystyle A_{n}}$

The alternating groups are subsets of the corresponding symmetrical groups . The alternating group is of particular importance . That it is the only non-trivial normal divisor of is an important part of the proof of the Abel-Ruffini theorem . This theorem from the beginning of the 19th century says that polynomial equations of the fifth or higher degree cannot be solved by root expressions . ${\displaystyle S_{n}}$ ${\displaystyle A_{5}}$ ${\displaystyle S_{5}}$

properties

The alternating groups are only defined for. ${\displaystyle n\geq 2}$

The alternating group consists of (half factorial ) elements. Only the groups and are Abelian . The alternating group is the commutator group of the symmetrical group . ${\displaystyle A_{n}}$ ${\displaystyle {\tfrac {n!}{2}}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{3}}$ ${\displaystyle A_{n}}$ ${\displaystyle S_{n}}$

Except for and , all alternating groups are simple . is the smallest nonabelian simple group; it is isomorphic to the rotation group of the icosahedron (see icosahedron group ). ${\displaystyle A_{2}}$ ${\displaystyle A_{4}}$ ${\displaystyle A_{5}}$

Generating system

The alternating group is of the 3-cycles of the symmetric group generated . ${\displaystyle A_{n}}$ ${\displaystyle S_{n}}$

Every 3-cycle is an even permutation because it is the product of two transpositions ${\displaystyle (a~b~c)}$

${\displaystyle (a~b)\circ (b~c)=(a~b~c)}$

can be written, and therefore an element of the alternating group. Furthermore, every even permutation is a product of 3-cycles, since pairs of two transpositions are products of 3-cycles. The following applies in detail

• ${\displaystyle (a~b)\circ (a~b)=\mathrm {id} =(a~b~c)\circ (c~b~a)}$if both transpositions are equal.
• ${\displaystyle (a~b)\circ (a~c)=(a~c~b)}$, if both transpositions have a common element.
• ${\displaystyle (a~b)\circ (c~d)=(a~c~b)\circ (a~c~d)}$if both transpositions have no element in common.

Inversions and number of inversions, even and odd permutations

One speaks of a misalignment or an inversion when two “places” of a permutation are in the “wrong” order. To determine the number of inversions of a permutation, all of its positions are compared with one another in pairs and the number of inversions is counted.

Example: The permutation in tuple notation has the inversions "3 before 1" and "3 before 2" (read from the two-line form ) and thus the inversion number . ${\displaystyle (3,1,2)}$ ${\displaystyle 2}$

One speaks of an even permutation if its inversion number is an even number; One speaks of an odd permutation if its inversion number is an odd number.

The signum is often defined as follows: ${\displaystyle \operatorname {sgn} \colon {\text{S}}_{n}\rightarrow \{+1,-1\}}$

${\displaystyle \operatorname {sgn} (p)=+1}$if the permutation is even, and ${\displaystyle p}$

${\displaystyle \operatorname {sgn} (p)=-1}$if is odd. ${\displaystyle p}$

The sign is a group homomorphism , so the following applies:

${\displaystyle \operatorname {sgn} (ps)=\operatorname {sgn} (p)\operatorname {sgn} (s)}$

for the permutations and . ${\displaystyle p}$ ${\displaystyle s}$

Group properties

The core of the sign is automatically a normal divisor of . You can also easily recalculate the subgroup properties: ${\displaystyle A_{n}}$ ${\displaystyle S_{n}}$

The following applies to the set of even permutations:

• The identical permutation is an element of this set. ${\displaystyle \mathrm {id} }$
• The set is closed in terms of concatenation , i.e. H. if and are even permutations, are also and even; a sketch of the evidence follows below. ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{1}\circ p_{2}}$ ${\displaystyle p_{1}^{-1}}$

With these requirements "inherits" all necessary group properties directly from : ${\displaystyle A_{n}}$ ${\displaystyle S_{n}}$

• For all even permutations the following applies: ${\displaystyle p_{1},p_{2},p_{3}\in A_{n}}$ ${\displaystyle p_{1}\circ \left(p_{2}\circ p_{3}\right)=\left(p_{1}\circ p_{2}\right)\circ p_{3}}$
• For all even permutations the following applies: ${\displaystyle p_{1}}$ ${\displaystyle p_{1}\circ \mathrm {id} =\mathrm {id} \circ p_{1}=p_{1}}$
• For all even permutations the following applies: There is an even with . ${\displaystyle p_{1}\in A_{n}}$ ${\displaystyle p_{1}^{-1}\in A_{n}}$ ${\displaystyle p_{1}\circ p_{1}^{-1}=p_{1}^{-1}\circ p_{1}=\mathrm {id} }$

The group is special because it is the smallest simple non-Abelian group. ${\displaystyle A_{5}}$

Seclusion

Transpositions

A transposition is a permutation in which exactly two different positions are exchanged, e.g. B. , where 3 and 5 are swapped. ${\displaystyle (5~3)}$

In general, the following applies to all -digit permutations and : can be generated with a finite number of transpositions from . ${\displaystyle n}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{1}}$

A special case of this applies to any permutation : can be generated from the identical permutation with a finite number of transpositions . ${\displaystyle p_{2}}$ ${\displaystyle p_{2}}$ ${\displaystyle \mathrm {id} }$

The picture shows how the permutation in tuple notation is generated from 5 transpositions. ${\displaystyle (2~5~3~1~4)}$ ${\displaystyle (1~2~3~4~5)}$

There is a certain amount of freedom in choosing the necessary transpositions; in the picture on the right, for example, the transpositions and could be omitted because they obviously cancel each other out. You could also increase the number of transpositions to 7, 9, 11, ... by adding further transpositions that cancel each other out in pairs. However, it is not possible to create an even number of transpositions from . ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle (2~5~3~1~4)}$ ${\displaystyle (1~2~3~4~5)}$

Transpositions and inversions

A single transposition always changes the value of the inversion number by an odd number, i.e. H. an even permutation becomes an odd one and vice versa.

When transposing from

${\displaystyle \left(\dotsc ,x,\dotsc ,y_{i},\dotsc ,z,\dotsc \right)}$ the new permutation

${\displaystyle \left(\dotsc ,z,\dotsc ,y_{i},\dotsc ,x,\dotsc \right)}$ generated,

the change in the inversion number is made up of the sum of the following changes:

• Change resulting from the new order of and , this is +1, if , otherwise −1. ${\displaystyle x}$ ${\displaystyle z}$ ${\displaystyle x<z}$
• Change resulting from the new order of and . ${\displaystyle x,y_{i}}$ ${\displaystyle z}$
  • If the largest or smallest element is of, the change is 0. ${\displaystyle y_{i}}$ ${\displaystyle x,y_{i},z}$
  • If is middle element of , the change is +2 or −2. ${\displaystyle y_{i}}$ ${\displaystyle x,y_{i},z}$

The sum of an odd and any number of even numbers always results in an odd number.

Conversion between even and odd permutations through transpositions

The statement made above can be generalized:

• With an odd number of transpositions, the value of the inversion number always changes by an odd number, i.e. H. an even permutation becomes an odd one and vice versa.
• With an even number of transpositions, the value of the inversion number always changes by an even number, i.e. H. an even permutation becomes an even permutation again and an odd permutation becomes an odd permutation again.

Transpositions and seclusion

Since id is an even permutation, we have:

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

If and are even permutations, then there are even numbers and , so that and can be represented as a chain of transpositions as follows: ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p_{n}}$ ${\displaystyle q_{n}}$ ${\displaystyle p}$ ${\displaystyle q}$

• ${\displaystyle p=t_{p_{1}}\circ \dotsb \circ t_{p_{n}}}$
• ${\displaystyle q=t_{q_{1}}\circ \dotsb \circ t_{q_{n}}}$

This means that the concatenation is also straight. ${\displaystyle p\circ q=t_{p_{1}}\circ \dotsb \circ t_{p_{n}}\circ t_{q_{1}}\circ \dotsb \circ t_{q_{n}}}$ ${\displaystyle p\circ q}$

Analogously one can derive: The concatenation of an even and an odd permutation always creates an odd permutation. The assumption that a permutation is even and odd leads to a contradiction. ${\displaystyle p}$ ${\displaystyle p^{-1}}$ ${\displaystyle p\circ p^{-1}=\mathrm {id} }$

Presentation of group A _n

A presentation by generators and relations looks like this: The group is for by ${\displaystyle A_{n}}$ ${\displaystyle n\geq 3}$

Generative and ${\displaystyle x_{1},\dotsc ,x_{n-2}}$

Relations

${\displaystyle x_{1}^{3}=x_{i}^{2}=e}$   For   ${\displaystyle 2\leq i\leq n-2}$

${\displaystyle (x_{i}x_{i+1})^{3}=e}$   For   ${\displaystyle 2\leq i\leq n-3}$

${\displaystyle (x_{i}x_{j})^{2}=e}$   For   ${\displaystyle 1\leq i\leq n-4,i+1<j}$

Are defined. This means that every group which contains elements which mutually satisfy the above equations and which together create the group is already isomorphic to the alternating group . ${\displaystyle n-2}$ ${\displaystyle x_{1},\dotsc ,x_{n-2}}$ ${\displaystyle A_{n}}$

This can be used to show that is isomorphic to the group of invertible matrices over the body with two elements. This follows from the fact that ${\displaystyle A_{8}}$ ${\displaystyle \mathrm {GL} _{4}(2)}$ ${\displaystyle 4\times 4}$

${\displaystyle x_{1}={\begin{pmatrix}1&1&1&1\\0&0&0&1\\1&1&0&0\\0&1&0&1\end{pmatrix}}\quad \quad x_{2}={\begin{pmatrix}0&1&0&1\\0&0&1&0\\0&1&0&0\\1&0&1&0\end{pmatrix}}\quad \quad x_{3}={\begin{pmatrix}0&1&1&1\\0&1&0&1\\1&1&0&0\\0&0&0&1\end{pmatrix}}}$

${\displaystyle x_{4}={\begin{pmatrix}1&0&1&0\\0&1&0&0\\0&0&1&0\\0&1&0&1\end{pmatrix}}\quad \quad x_{5}={\begin{pmatrix}0&0&1&0\\0&1&0&1\\1&0&0&0\\0&0&0&1\end{pmatrix}}\quad \quad x_{6}={\begin{pmatrix}0&1&1&1\\0&0&1&0\\0&1&0&0\\1&1&1&0\end{pmatrix}}}$

create the group and satisfy the above relations.

See also

• alternating group of grade 4
• alternating group of grade 5

literature

• Christian Karpfinger, Kurt Meyberg: Algebra. Groups - rings - bodies. Spektrum Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2018-3 , pp. 108-109

Individual evidence

1. ^ B. Huppert: Endliche Gruppen I , Springer-Verlag (1967), Chapter I, Theorem 6.14
2. ↑ B. Huppert: Endliche Gruppen I , Springer-Verlag (1967), Chapter II, Sentence 2.5