Sylow $p$-subgroups of Galois group

Question

It just my curious, but I couldn’t find any related concept:

Condition)

Let $G$ be a finite Galois group, and the number of each Sylow $p_{i}$-subgroup of $G$ is one, where $p_{i}$ is a prime factor of $|G|$.

Question)

Under the above condition, is it true $G$ is cyclic?

Give some advice or related notion! Thank you!

Answer 1

Every finite group is a Galois group of some finite extension $L/K$ of fields.

If a finite group has a unique Sylow subgroup for each prime, then all one can conclude is that it is nilpotent, that is a direct product of $p$-groups. It need not be cyclic, or indeed Abelian.