Let G be a group and g be an element of G. The smallest positive integer...

 

Transcribed Image Text:

Let G be a group and g be an element of G. The smallest positive integer n such that g" = e is called the order of g. If such integer does not exist, then we say that g has infinite order. The order of g is denoted by g a) Assume that [g] =6. Show that g|-6 and |g³|= 2. b) More generally, if [g] =n, then for any nonzero integer k there is lg*1 = (mk) where (n, k) denote the greatest common divisor of n and k.. e) Show that the cyclic subgroup (g) of G, generated by the element g, has the order 19. Let G be a group and g be an element of G. The smallest positive integer n such that g" = e is called the order of g. If such integer does not exist, then we say that g has infinite order. The order of g is denoted by g a) Assume that [g] =6. Show that g|-6 and |g³|= 2. b) More generally, if [g] =n, then for any nonzero integer k there is lg*1 = (mk) where (n, k) denote the greatest common divisor of n and k.. e) Show that the cyclic subgroup (g) of G, generated by the element g, has the order 19.