A group consists of a set of elements (objects, quantities) finite or infinite in number with a rule for combining the elements ( multiplication rule) such that the set is closed under multiplication. Thus, if G represents a group, with ( g ₁, g ₂, , g _n) as its elements, then the combination of any two of its elements g _i and g _j, denoted by g _i g _j also belongs to the group. (Mathematically, g ? G stands for the statement g belongs to G.). It should be understood that the combination rule for the elements (namely, the multiplication rule) does not necessarily have to be an ordinary product of the elements. It can also be any other operation such as addition.

The set of elements have to satisfy several other properties in order to define a group, and these are

1. The multiplication (combination) of the elements must be associative, namely,

   (D.1)

2. There must be an identity element of the group, denoted as I, such that combining any element with the identity gives back the same element,

   (D.2)

3. For every element g ? G, there must exist a unique inverse element g ^?1 ? G, such that

   (D.3)

For a simple example of a group, let us assume that G consists of all the real numbers, both positive and negative. In this case, we can define

(D.4)

With this combination formula,...