Group with Cyclic Sylow Subgroups

Proposition

Let $G$ be a group such that each Sylow subgroup is cyclic. Then $G$ is solvable. Furthermore, $G$ is metacyclic, i.e., $G/C$ is cyclic and $C$ is cyclic.

\begin{proof} By Burnside’s transfer theorem, $G$ is solvable. By the property of Fitting subgroups we obtain that $F:=\mathrm{F}(G)$ is a cyclic group and so $C_G(F)=F$. Then by NC Lemma, $G/C_G(F)\cong N_G(F)/C_G(F)\lesssim \mathrm{Aut}(F)$ is abelian. Now we finish the proof. \end{proof}