Question

Find all the generators of each of the following cyclic groups
a. The even numbers under addition

b.

{Z20, +}

c.

d.

Answer 1

1 b7 {20. this a finite cyclic group order of this group is 20. ... the Numbers which are prime to 20 is given by 1,3,7, 11, 13, 17, 19. [less than 20]. Therefore, the generators of the cyclic group face { Z20-t} is = 1, 3, 7, 11, 13, 17, 1j c) G=<a) where lal = 15. ... Now, G=(a) be a cyclic group of order to 15 we know that an is a generator of G iff to is Relatively prime to 15. Now, the numbers which are less than 15 and prime to 15 is given by, 1, 2, 4, 7, 8, 11, 13, 14, This G=<a> has 8 generators which are a, az, as, at, as, all, a3, al4. d) G=<a37 and o(a) = 15 . Now, G=(a) and o(61) = 15 = 0(a). .. Now, o (23) = 15 = 15 = 5 .: thus o(a) = 5. :. Now, the numbers which are less than 5 and prime to 5 is given by 1, 2, 3, 4. - Thus, the generators of G, is given by, {as, as, as, a2}

a) Now, (22, +).=G. since G is an Infinite cyclic group. Now, we know that An infinite cyclic group can have only two generators. so, the generators of G is soos are 2,(-2).