Question

What is the necessary and sufficient condition for an integer k to be a generator of Zn?

a) gcd(n,k) = 1.

b) k divides n.

c) n is some power of k.

d) n=k.​​

Answer 1

a) gcd (n, k) = 1

The necessary and sufficient condition for an integer k to be a generator of Zₙ is gcd (n, k) = 1.

Explanation:

• We have to remember that a generator generates the complete cyclic group, i.e., all its elements.

• Examples of some cyclic groups are Z₆, Z₈ and Z₂₀. They have one common generator 1.

• Cyclic groups can be finite or infinite. Every cyclic group is abelian.