Totient euler function: Why $\gcd(n,k)=1$ is important for $\zeta _n^k$.

Question

Let $\mu_n=\{\zeta \mid \zeta^n-1=0\}=\{1,\zeta _n,\zeta _n^2,...,\zeta _n^{n-1}\}$. We call generator an element $\zeta_n^k$ when $\gcd(k,n)=1$. Why those number are such important ? I think that $\left<\zeta _n^k\right>$ is a subgroup of $\left<\zeta _n\right>$ if and only if $(k,n)=1$, and that if $\gcd(k,k')=1$ $\left<\zeta _n^k\right>\cap \left<\zeta _n^{k'}\right>=\{1\}$, but I still not see the information it gives.