Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $P\subseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple.
I understand the pieces of information given in the problem but how to put them together to make an argument is challenging to me. I will appreciate any help.
I am thinking that, if I start by assuming $G$ is not simple and that there exists a normal nontrivial subgroup $ N\triangleleft G$.
And if I can show that the commutator subgroup $G'=1$ then since $G'=G$ I get a contradiction to the nontriviality assumption of the group.
Is this a good way to go? How should I view it? Thanks