# Examples of Abelian Groups

Examples of Abelian Groups

In an Abelian group, the elements commute: for any elements and in the group. Therefore, the multiplication table of an Abelian group is symmetric, and plotting the resulting pattern can reveal interesting structures. The index represents the direct product of the cyclic groups ⊗⊗…. 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.

AB=BA

A

B

{,,…}

n

1

n

2

C

n

1

C

n

2