How do I find a primitive element in $Z_7$?

Question

I understand the definition of a primitive element. I also know that 3 is primitive element of $Z_7$. I was shown that 2 is not primitive element of $Z_7$ because $2^{3}$ = 8= 1. I do not understand why 2 is not a primitive element because of that reason. Can someone explain please?

Answer 1

It's because a primitive root, k, modulo n is a number such that you can write every number in your modulus as a power of k.

So, here, you can write every element of Z₇ as a power of 3 (various powers of 3 in modulo 7, in order, are 3, 2, 6, 4, 5, 1, at which point it loops) where you cannot write everything as a power of 2 because it loops at 2, 4, 1, 2, 4, 1.

In particular, the rigorous defition of a primitive root, k, modulo n is that, for every integer, j, coprime to k (ie they share no common factors but 1) there exists a power of k such that said power is congruent to j.

Hope that helps!