Integral power of an element Algebraic structures
(Monday, May 3, 2010)
Let us consider the product of n elements each equal to a in a multiplicative group G. By the general associative law this product is independent of the manner of parenthesizing so that there exists a unique value of the product which we denote by the conventional symbol an, where n is a positive integer.
If n is negative integer, let n = -m, where m is a positive integer. Then we shall write a-m = (am)-1.
We have (a.a.a…a) -1 = a-1.a-1.a-1…a-1
(m times) (m times)
Since (a1 a2 …ak) -1 = ak-1 ak-1-1…a2-1a1-1
\ (am)-1 = (a-1)m, so that a-m = (a-1)m
Also, we write a0 = e.
Hence, the symbol an is well-meaningful for every integral value of n (positive, negative or zero).
It may now be easily proved that
am+n = am.an and (am) n = amn for all a e G and m, n e Z.
The least positive integer n such that an = e.a e G with identity element e is called the order of the element a of the group G.
Posted in Posted by waytofeed at 7:30 AM
0 comments:
Post a Comment