Finite field generator exponentiation properties

Question

For a finite integer field $\mathbb{Z}_n$ with a generator $g$ and $x\in\mathbb{Z}_n$, a property I do not understand arises and I have not come across an explanation for it. If another set of values $x_1=n+x,\ x_2=2n+x\dots$ then $g^{x}\neq g^{x_1}\neq g^{x_2}\dots$. Why is this? I've noticed and conjectured that for an $x_k=kn+x$, $g^{x}=g^{x_k +k}$ at least for several examples I have been toying with to get a feel for the mechanics. Since all these values are members of the same congruence class, I'd expect them to generate identical field elements. Is it because the multiplicative subgroup is of order $n-1 $ due to the removal of the additive identity? I'd appreciate any guidance and clarification on this matter.

Answer 1

If your $n$ is a prime number, by lil' Fermat, we have $g^n\equiv g\mod n$, hence $$g^{x_k}=g^{x+k}\enspace\forall k\in\mathbf Z.$$