# 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:

AbCn1Cn2Cnm

## 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

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:

• 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.