Maths - Alternating Groups

The alternating group is important from a mathematical point of view because, for A5 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:

AnSnC2

Sn is the semidirect product An&22CA;C2

Example for A3

If we start with S3 then there are 6 permutations:

element
(permutation)
i a b c d e
S3 i a b c d e

This contains three 2-cycles as follows:

element
(permutation)
a b e
2-cycles a b e

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 i   0 even
a a a 1 odd
b b b 1 odd
c c ab 2 even
d d ba 2 even
e e 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                  
S1           ()        
S2         () (1,2)      
S3 A3     () (2,3)
(1,3)
(1,2)
(1,2,3)
(1,3,2)
   
S4 A4   ()

(3,4)
(2,4)
(1,4)
(2,3)
(1,3)
(1,2)

(1,2),(3,4)
(2,3),(1,4)
(1,3),(2,4)

(1,3,4)
(1,4,3)
(2,3,4)
(2,4,3)
(3,1,2)
(3,2,1)
(4,1,2)
(4,2,1)

(1,2,3,4)
(1,2,4,3)
(1,3,2,4)
(1,3,4,2)
(1,4,2,3)
(1,4,3,2)
 
S5 A5 ()

(4,5)
(3,5)
(2,5)
(1,5)
(3,4)
(2,4)
(1,4)
(2,3)
(1,3)
(1,2)

(1,2),(3,4)
(1,2),(4,5)
(1,2),(3,5)
(2,3),(1,4)
(2,3),(1,5)
(2,3),(5,4)
(1,3),(2,4)
(1,3),(2,5)
(1,3),(5,4)
(),()

(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 An cycle notation Ancycle compressed Signature
2   <1> 12
3   <(1 2 3)> 13,3
4 <(), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)> <(1 2)(3 4),(1 2 3)> 14,3
5   <(3 4 5),(1 2 3)> 15,(22,1),(3,12),(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)


a a
a a
 

                                                           Type: Table(2)
permutationRepresentation(a1,1) 


[
1
]
                      
                           
                                              Type: List(Matrix(Integer))
a2 := alternatingGroup(2)
   

<1>
 
      
                                          Type: PermutationGroup(Integer)
toTable()$toFiniteGroup(a2,1)
  

a a
a a
                      
                           
                                                           Type: Table(2)
permutationRepresentation(a2,2)


[
1 0
0 1
]
                      
                           
                                              Type: List(Matrix(Integer))
a3 := alternatingGroup(3)


<(1 2 3)>
    
         
                                          Type: PermutationGroup(Integer)
toTable()$toFiniteGroup(a3,1)


i a aa
a aa i
aa i a
                      
                           
                                                           Type: Table(3)
permutationRepresentation(a3,3)


[
0 0 1
1 0 0
0 1 0
]
                      
                           
                                              Type: List(Matrix(Integer))
a4 := alternatingGroup(4)


<(1 2)(3 4),(1 2 3)>
               
                    
                                          Type: PermutationGroup(Integer)
toTable()$toFiniteGroup(a4,1)
    

i a b ab ba bb aba abb bab bba babb bbab
a i ab b aba abb ba bb bba bab bbab babb
b ba bb bab bba i abb babb bbab a aba ab
ab aba abb bba bab a bb bbab babb i ba b
ba b bab bb abb babb bba i a bbab ab aba
bb bba i bbab a b babb aba ab ba abb bab
aba ab bba abb bb bbab bab a i babb b ba
abb bab a babb i ab bbab ba b aba bb bba
bab abb babb a bbab ba i ab aba b bba bb
bba bb bbab i babb aba a b ba ab bab abb
babb bbab ba aba b bab ab bba bb abb i a
bbab babb aba ba ab bba b bab abb bb a i
                      
                           
                                                          Type: Table(12)
permutationRepresentation(a4,4)


[
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
,
0 0 1 0
1 0 0 0
0 1 0 0
0 0 0 1
]
                      
                           
                                             Type: List(Matrix(Integer))
a5 := alternatingGroup(5)
   

<(3 4 5),(1 2 3)>
            
                 
                                          Type: PermutationGroup(Integer)
order(a5)


60
 
      
                                                    Type: PositiveInteger
permutationRepresentation(a5,5)


[
1 0 0 0 0
0 1 0 0 0
0 0 0 0 1
0 0 1 0 0
0 0 0 1 0
,
0 0 1 0 0
1 0 0 0 0
0 1 0 0 0
0 0 0 1 0
0 0 0 0 1
]
                      
                           
                                              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)

[
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
,
0 0 1 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
]

                                              Type: List(Matrix(Integer))
(19) ->

where:


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see also:

 

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