(1). G is is finite implies o(G) is finite.
Let G be a finite group of order n and let e be the identity element in G. Then the elements of G may be written as e, g1, g2, ... gn-1. We prove that the order of each element is finite, thereby proving that G is finite implies that each element in G has finite order. Let gkbe an element in G which does not have a finite order. Since (gk)r is in G for each value of r = 0, 1, 2, ... then we conclude that we may find p, q positive integers such that (gk)p = (gk)q . Without loss of generality we may assume that p> q. Hence
(gk)p-q = e. Thus p - q is the order of gk in G and is finite.
(2). o(G) is finite implies G is finite.
This follows from the definition of order of a group, that is, the order of a group is the number of members which the underlying set contains. In defining the order we are hence assuming that G is finite. Otherwise we cannot speak about quantity.
Hope that this helps.
numerical order
rank
range
range
It is ordered data.
numerical order
Data in order from smallest to largest or vice-versa is called numerical order. It is a systematic arrangement of numbers.
Because the bad people of the world will always need someone to prove themselves on -and vice versa!
range
rank
range
Ordered data.
The data are sorted.
Range what about ranking?
It is ordered data.
It is ordered data.
The phrase Vice Versa is Latin for the reverse order from the way something has been stated. This was popular from 1595 to 1605.