Problem: If $G$ is a finite group whose Sylow $p$-subgroups are all cyclic then $G$ has normal subgroup $N$ and such that $G/N$ and $N$ are both cyclic.
Whenever I need to find normal subgroup, I always try to find one homomorphim $\phi$ to show $\text{ker}\:\phi$ is non-trivial for a homomorphism $\phi$ on $G$.
But for this problem, I couldn't come up with any trivial one. I guess some tricky observation is needed. Any hint will be appreciated.
This is a result of Hölder, Burnside, and Zassenhaus. See 10.1.10 in Robinson "A Course in the Theory of Groups". The proof uses various tools such as Burnside's transfer theorem that are rather more sophisticated than just the Sylow theorems, so this "exercise" does not seem appropriate to your level of knowledge.
As a special case, if $G$ is simple and nonabelian, the exercise implies that $G$ has a non-cyclic Sylow subgroup. How are you supposed to prove this using just the Sylow theorems?