Suppose is a group and the composition has been denoted by multiplicatively, let . Then by closure property etc., are all elements of . Since the composition in obeys general associative law, therefore to factors is dependent of the manner in which the factors may be grouped.
If is a positive integer, we define factors to factors. Obviously . If is the identity element of the group , then we define .
If is a positive integer then is a negative integer. Now we define where is the inverse of in . Thus, . Thus we have defined for all integral values of positive, zero or negative.
Integral Multiples of an Element of a Group
If in a group the composition has been denoted additively, then in place of using the word integral powers of an element of a group we use the word integral multiples of an element of a group. The difference is only of notation, otherwise the meaning is the same. Thus in this case if is a positive integer we write in place of and we define up to terms.
In place of we write . Thus we define where is the identity of .
If is a positive integer, then in place of we write .
Thus, we define where denotes the inverse of in .
In multiplicative notation the following laws of indices can be easily proved:
and where is the set of integers.
In additive notation the following laws of multiples can be easily proved: