Prove the following: A group Gof order nis cyclic if and only if for every divisor

dof nthere are at most ? ' (d ) elements of order din G(hint: use the previous

exercise; ?'is the Euler phi-function).

previous exercise for reference?

45. Show that a nite cyclic group of order n has exactly one subgroup of each order :1 dividing it, and that these are all the subgroups it has. 46. The Euler phi-functiou is dened for positive integers n by 9001) = s, Where s is the number of positive integers less than or equal to n that are relatively prime to :1. Use Exercise 45 to Show that n = 290M), aim the sum being taken over all positive integers d dividing n. [Hint Note that the number of generators of Ed is (Md) by Corollary 6.16.]