The alternating group is important from a mathematical point of view because, for A_{5} and above, it is a simple group which means it cannot be factored into smaller groups. It therefore plays an important pat in the categorization of groups.
The alternating group is a group containing only even permutations of the symmetric group. It turns out that half the permutations of the symmetric group are even and the other half are odd. That is if we permute a set with 'n' elements the symmetric group has !n permutations and the alternating group has !n/2 permutations.
Even and Odd Permutations
Every permutation in a symmetric group (in other words, for every group) can be expressed as a product of 2-cycles. If the permutation has an even number of 2-cycles then it is an even permutation and if the permutation has an odd number of 2-cycles then it is an odd permutation.
In technical terms there is a short exact sequence:
A_{n}→S_{n}→C_{2}
S_{n }is the semidirect product A_{n}C_{2}
Example for A_{3}
If we start with S_{3} then there are 6 permutations:
element (permutation) |
i | a | b | c | d | e |
---|---|---|---|---|---|---|
S_{3} |
This contains three 2-cycles as follows:
element (permutation) |
a | b | e |
---|---|---|---|
2-cycles |
So we can construct any permutation element from a sequence of 2-cycles like this:
permutation | sequence of 2-cycles | number of 2-cycles | even or odd | |
---|---|---|---|---|
i | 0 | even | ||
a | 1 | odd | ||
b | 1 | odd | ||
c | 2 | even | ||
d | 2 | even | ||
e | 1 | odd |
Size of Alternating sets
number of elements of a set | possible ways to order set |Sn| | possible ways to order set |An|= |Sn| /2 |
---|---|---|
1 | 1 | |
2 | 2 | 1 |
3 | 6 | 3 |
4 | 24 | 12 |
n | !n | !n/2 |
Signature or Cycle Shape
symmetric group | alternating group | ||||||||||
S_{1} | () | ||||||||||
S_{2} | () | (1,2) | |||||||||
S_{3} | A_{3} | () | (2,3) (1,3) (1,2) |
(1,2,3) (1,3,2) |
|||||||
S_{4} | A_{4} | () | (3,4) (2,3),(1,4) (1,3),(2,4) |
(1,3,4) |
(1,2,3,4) (1,2,4,3) (1,3,2,4) (1,3,4,2) (1,4,2,3) (1,4,3,2) |
||||||
S_{5} | A_{5} | () | (4,5) (1,2),(3,4) |
(2,3,4) (2,4,3) (1,3,4) (1,4,3) (4,1,2) (4,2,1) (3,1,2) (3,2,1) |
(1,2,3,4) (1,2,4,3) (1,3,2,4) (1,3,4,2) (1,4,2,3) (1,4,3,2) |
(1,2,3,4,5) | |||||
n | A_{n} cycle notation | A_{n}cycle compressed | Signature |
---|---|---|---|
2 | <1> | 1^{2} | |
3 | <(1 2 3)> | 1^{3},3 | |
4 | <(), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)> | <(1 2)(3 4),(1 2 3)> | 1^{4},3 |
5 | <(3 4 5),(1 2 3)> | 1^{5},(2^{2},1),(3,1^{2}),(5) | |
6 | <(1 2)(3 4 5 6),(1 2 3)> | ||
7 | <(3 4 5 6 7),(1 2 3)> | ||
8 | <(1 2)(3 4 5 6 7 8),(1 2 3)> | ||
9 | <(3 4 5 6 7 8 9),(1 2 3)> | ||
10 | <(1 2)(3 4 5 6 7 8 9 10),(1 2 3)> |
Generating a Alternating Group using a Program
We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.
a1 := alternatingGroup(1)<1> Type: PermutationGroup(Integer) toTable()$toFiniteGroup(a1,1)
Type: Table(2) permutationRepresentation(a1,1)
Type: List(Matrix(Integer)) a2 := alternatingGroup(2)<1> Type: PermutationGroup(Integer) toTable()$toFiniteGroup(a2,1)
Type: Table(2) permutationRepresentation(a2,2)
Type: List(Matrix(Integer)) a3 := alternatingGroup(3)<(1 2 3)> Type: PermutationGroup(Integer) toTable()$toFiniteGroup(a3,1)
Type: Table(3) permutationRepresentation(a3,3)
Type: List(Matrix(Integer)) a4 := alternatingGroup(4)<(1 2)(3 4),(1 2 3)> Type: PermutationGroup(Integer) toTable()$toFiniteGroup(a4,1)
Type: Table(12) permutationRepresentation(a4,4)
Type: List(Matrix(Integer)) a5 := alternatingGroup(5)<(3 4 5),(1 2 3)> Type: PermutationGroup(Integer) order(a5)60 Type: PositiveInteger permutationRepresentation(a5,5)
Type: List(Matrix(Integer)) a6 := alternatingGroup(6)<(1 2)(3 4 5 6),(1 2 3)> Type: PermutationGroup(Integer) order(a6)360 Type: PositiveInteger permutationRepresentation(a6,6)
Type: List(Matrix(Integer)) (19) -> |
where:
- The points of the permutation are numbered 1..n
- The elements of the group are named: "i" for the identity, single letters "a","b"... for the generators, and products of these.
- numbers in brackets are points of permutations represented in cyclic notation.
- The permutation is represented by a set of comma seperated permutations in angle brackets like this: <(1 2)(3 4),(1 2 3)>
- non-changing elements of the permutation are ommited so the above case is equivalent to: <(1 2)(3 4),(1 2 3)(4)>
- attempting to produce a alternating group with less than 3 elements does not produce a valid result.
- The Axiom/FriCAS program can't work in terms of the Cayley table, so I have added my own code to do this.