Characterizing Generators of $\mathbb{Z}_p^*$

Question

I have started to study cyclic groups and generator. I can't prove that $g\in \mathbb{Z}_{p}^{*}$ is a generator for cyclic group $\mathbb{Z}_{p}^{*}$ if and only if $g^{\frac{p-1}{q_{i}}}\neq 1\;\forall i=\overline{1,r}$, where $p$ is prime, $p>2$, and $q_{1},q_{2},...q_{r}$ prime facotrs for $p-1$.

Answer 1

I think your statement makes it look a little bit more confusing than it actually is. Take any cyclic group G of order n and an element $g\in G$. Then we know by Lagrange that $ord(g) | n$. Clearly, if $g$ generator, then $g^{n/q} \neq 1$ for any $q|n$ (any $q$ that divides $n$) different than 1.

Conversely, if $g$ not generator, then $g^{n/q}=1$ for some $q|n$. If q is prime, you 're done, if not, and if say $m$ is a prime factor of $q$ and $q=ms$ then $g^{q/m}=(g^{n/q})^s=1$ which is what you wanted.