I had tried some questions in Group Theory in January but Could not post the questions on which I am struck because of my illness. So, I am writing them now. I have done a graduate level course on group theory.
Prove that $S_n$ is solvable for $n\leq4$, but $S_3$ and $S_4$ are not nilpotent.
It is trivial for $n \leq 2$. If I use the definition of derived subgroup for $n=3$, $n=4$ and proving $G'_n=\langle e\rangle$, then it will be a brute force method and I don't feel like doing it. So, I am looking for some elegent way.
Also, I am at loss of ideas on how to prove that $S_3$ and $S_4$ are not nilpotent. Actually, It is my weak point. So, can you also give some hint for this. Thanks!
A nilpotent group has a nontrivial center, and the center of group $S_3$ (and $S_4$) is trivial.