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 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: