Maths - Abelian Groups

Abelian Groups are groups which commute (ab = ba)

The Fundamental Theorem of Abelian Groups

Every finite abelian group 'Ab' is isomorphic to a direct product of cyclic groups, That is:

Ab≡Cn1⊗Cn2⊗⊗Cnm

Cyclic Groups

The properties of cyclic groups are discussed on this page and listed on this datasheet so we will just summarise here:

Cayley Table Cayley Graph Cyclic Notation
  1 a b c ... n
1 1 a b c ... n
a a b c ... n n-1
b b c ... n n-1  
c c ... n n-1    
... ... n n-1      
n n n-1        
cyclic cayley graph (1,2,3..n)
Group Presentation Group Representation  
<a | an=1>
0 0 0 0 1
1 0 0 0 0
0 1 0 0 0
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
0 0 1 0 0
0 0 0 1 0
 

Products of Cyclic Groups

Commutivity in Caley Diagrams

commutative
C3⊗C2

non-commutative
D3
c3c2 d3

Commutivity in Multiplication Tables

a
1 2
2 1
b
1 2
2 1
c
1 2
2 1
b
1 2
2 1
c
1 2
2 1
a
1 2
2 1
c
1 2
2 1
a
1 2
2 1
b
1 2
2 1

Commutivity in Cyclic Notation

If we are multiplying a 3 element cycle by a two element cycle we number the elements in the 2×3 rectangle:

1 2 3
4 5 6

we then create 2 permutations, one from the rows and the other from the columns:

<(1 2 3)(4 5 6),(1 4)(2 5)(3 6)>

Simarly for n×m multipication:

1 2 n n+1 )
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
1 2 n n+1 )

Generating a Abelian Group using a Program

We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.

(1) -> )r axiom/abelian
)set output algebra off
                       
)set output mathml on  
                       
C2 := FiniteGroup(2,[[1,2],[2,1]],["a","b"])

                                                               Type: Type
AB6 := directProduct([[1,2,3],[2,3,1],[3,1,2]],["1","2","3"])$C2

                                                               Type: Type
toTable()$AB6
a1 a2 a3 b1 b2 b3
a2 a3 a1 b2 b3 b1
a3 a1 a2 b3 b1 b2
b1 b2 b3 a1 a2 a3
b2 b3 b1 a2 a3 a1
b3 b1 b2 a3 a1 a2
                                                           Type: Table(6)
setGenerators([false,true,false,true,false,false])$AB6
                                                              Type: Void
PAB6 := toPermutation()$AB6
<(1 2 3)(4 5 6),(1 4)(2 5)(3 6)>
                                          Type: PermutationGroup(Integer)
permutationRepresentation(PAB6,6)
[
0 0 1 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 0 0 1
0 0 0 1 0 0
0 0 0 0 1 0
,
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
]
                                              Type: List(Matrix(Integer))
(7) ->

where:

 


metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

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Terminology and Notation

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