If a groupis abelian thenfor all

This has several consequences.

All cyclic groups are abelian since a cyclic group is generated by a single element, somay be writtenwithThe group is abelian since ifthenAn abelian group may always be constructed in this way.

The group table is symmetric about the main diagonal since the element in row i, column j, written so the elements in positionsandare identical.

If the group table is symmetric then the group is abelian sincefor all

There are some results connected with these two results.

Any group with a prime number of elements is abelian. It is generated by a single element, the powers of which must commute.

**Examples of Abelian Groups**

All rotation groups in the plane are abelian.

The integers under addition or multiplication.

Addition or multiplication modulo n (if a group is present).

The real numbers under addition or multiplication.

Matrices under addition

Complex ij numbers under multiplication or addition

Hamiltonian ijk numbers under addition.

The Klein Group consisting of the group of symmetries of the rectangle.

**Examples of Non Abelian Groups**

Rotation groups in more than two dimensions.

The real numbers under addition or multiplication.

The group of invertible matrices under multiplication

The non zero Hamiltonian ijk numbers under multiplication.

The general dihedral groups consisting of the group of symmetries of regular polygons.

The general symmetric group S-n consisting of all permutations of the numbers