Examples of Abelian Groups

group

{4,3,3,3}

view

2D

3D

Cayley graph

In an Abelian group, the elements commute:

AB=BA

for any elements

A

and

B

in the group. Therefore, the multiplication table of an Abelian group is symmetric, and plotting the resulting pattern can reveal interesting structures. The index

{

n

1

,

n

2

,…}

represents the direct product of the cyclic groups

C

n

1

⊗

C

n

2

⊗…

. The Cayley graph encodes how the products of the group generators produce all the elements of the group (mouse over the vertices). A selection is shown because some examples are trivial, some combinations are not allowed, and tables grow arbitrarily large.