Some properties of a finite group with all Sylow subgroups that are cyclic

Question

I consider a finite group $G$ such that all its Sylow's subgroups are cyclic. I suppose that $|G|=p_1^{k_1}...p_n^{k_n}$ with $p_1<...<p_n$ distinct primes.

Can I say something about the normality of the $p_i$-Sylow subgroups? Can I say for instance that $P_n$ is normal?

I know then that if $P_1$ is a $p_1$- Sylow subgroup than $G$ has a $p_1$ normal complement K.

Is it true that $K$ is cyclic? How can I show that?

Thanks for the help!

Answer 1

In addition to the remarks of Tobias (with $G=S_3$ you can refute your statement), one can prove that if a group has cyclic Sylow subgroups, it must be solvable (of derived length $\leq 2$). For a proof see for example M.I. Isaacs, Finite Group Theory, Corollary 5.15