1) Define phi(n) to be the number of positive integers less than or equal to n a
ID: 2942102 • Letter: 1
Question
1) Define phi(n) to be the number of positive integers less than or equal to n and relatively prime to n. Use the fact that a cyclic group of order n has exactly phi(d) elements of order d for each d dividing n, and no elements of order d for each d not dividing n. Deduce that for any positive integer n, the sum of phi(d) over all d dividing n is in fact n.2) Now let G be a finite group of order n with at most one subgroup of any order. Prove G is cyclic. You may use the fact that the order of any element in a finite group must divide the order of the group.
Hint: Let N(d) be the number of elements in G of order d, and note that the above fact implies that N(d) is 0 if d does not divide n. Use this to show that n equals the sum of the N(d) over all d dividing n and explain why N(d) equals 0 or phi(d) for any such d. Compare this equation with the result in part a relating n to the sum of the phi(d), and deduce that N(d) must be nonzero for all d dividing n, so that in particular, N(n) is nonzero. Explain why this implies that G is cyclic.
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