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:

AbC_n1C_n2…C_nm

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		(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 C3C2	non-commutative D3

Commutivity in Multiplication Tables

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:

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

                                                           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)

                                              Type: List(Matrix(Integer))
(7) ->

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.
• abelianGroup[1] is not really valid and the results for this case are nonsense.
• The Axiom/FriCAS program can't work in terms of the Cayley table, so I have added my own code to do this.

EuclideanSpace

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