Question

Part 15A and 15B

(15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are relatively prime to d. Divisor Number of elements Number of integers between 1 and d relatively prime to d of order d 3 6 12 (b) Part (a) appears to indicate that there is a relationship between the number of elements in Zn of a given order and the number of integers relatively prime to that order. There is a useful function that describes this number. The Euler phi function ф is defined to give the number of positive integers less than or equal to and relatively prime to a given positive integer. For example, φ(2)-1 since l is the only positive integer less than 2 and relatively prime to 2. Similarly, p(3)-2, ф(4-2, yo(5-4, etc. Show that the number of elements in Z of order d is p(d)

Answer 1

a) Z₁₂={$\bar{0}$,$\bar{1}$,$\bar{2}$,$\bar{3}$,$\bar{4}$,$\bar{5}$,$\bar{6}$,$\bar{7}$,$\bar{8}$,$\bar{9}$,$\bar{10}$,$\bar{11}$}

Now, No. of elements of order 1=1($\bar{0}$)

No. of elements of order 2=1($\bar{6}$)

No. of elements of order 3=2($\bar{4}$,$\bar{8}$)

No. of elements of order 4 =2($\bar{3}$,$\bar{9}$)

No. of elements of order 6=2($\bar{2}$,$\bar{10}$)

No. of elements of order 12=4($\bar{1}$,$\bar{5}$,$\bar{7}$,$\bar{11}$)

Divisor d	Number of elements of order d	Number of integers between 1 and d relatively prime to d
1	1	1
2	1	1
3	2	2
4	2	2
6	2	2
12	4	4

b)We know that Z_n has exactly one subgroup of order d--call it <a>.Then every element of order d also generates the subgroup <a> since every element of order d generates a subgroup of order d and since Z_n is cyclic it has exactly one subgroup of order d.So all the elements of order d generates the same subgroup.Now <a> is a cyclic group since subgroup of a cyclic group is cyclic.In a cyclic group if a is a generator, then a^k is also a generator of that group if and only if gcd(k,n)=1.Number of such elements is precisely $\phi$(d).

So the number of elements of order d in Z_n is $\phi$(n)..........(Proved).