After searching through the online databases of number fields, I guess the following statements should be true, but I can not prove it.
Let $L/\mathbb{Q}$ be a cyclic Galois extension of prime degree $q$. Let $p\neq q$ be another prime. My Major guess is:
Major question: If $\mathcal{Cl(O}_L) \cong \dfrac{\mathbb{Z}}{p\mathbb{Z}}$, then $p\stackrel{q}{\equiv} 1$.
If the first statement is true, then I believe its proof would be very clarifying and enlightening for other statements. The second generalized version of the major question is:
If $\mathcal{Cl(O}_L) \cong (\dfrac{\mathbb{Z}}{p\mathbb{Z}})^f$, then $p^f\stackrel{q}{\equiv} 1$.
Also, I have some doubts about the third generalized version of that question:
If $\mathcal{Cl(O}_L) \cong (\dfrac{\mathbb{Z}}{p^{f'}\mathbb{Z}})^{f''}$, then $p^{f'f''}\stackrel{q}{\equiv} 1$.
I strongly doubt the correctness of this last statement, and I think this statement is incorrect.
Minor question: If $p < q$ and $ \mid \mathcal{Cl(O}_L) \mid = p^f$, then $\mid \mathcal{Cl(O}_L) \mid \stackrel{q}{\equiv} 1$, and $\mathcal{Cl(O}_L) \cong (\dfrac{\mathbb{Z}}{p^{f'}\mathbb{Z}})^{f''}$, with $f'f''=f$.