Abelian and Non Abelian Groups
If a group
is abelian then
for all
This has several consequences.
All cyclic groups are abelian since a cyclic group is generated by a single element, somay be written
with
The group is abelian since if
then
An 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 positions
and
are 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