How do you prove that order of a group G is finite only if G is finite and vice versa?

Answer 1

(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.