
A group is defined as a collection of elements (a set), which are labelled a, b, c, ... and so on, and which are related to one another by the following rules:
 If a and b are both members of the group G, then their product, c = ab is also a member of the group G.
 The process is associative, i.e., a(bc) = (ab)c.
 There must be an element, called the unit element and usually denoted by e, defined so that ae = a, be = b, and so on for all elements in the group.
 Each element has an inverse, written as a^{1}, b^{1} and so on, defined so that
aa^{1} = e and so on.


A group for which ab = ba is an Abelian group. The set of ordinary integer numbers ( ... 3, 2, 1, 0, 1, 2, 3, ...) under "addition" is a simple example of an Abelian group, where "0" is the unit element and the inverse is the same number with opposite sign. 