I would like to show that if$\ G $ is a finite group of order$\ p_1 p_2 ... p_k $ where$\ p_i $ are distinct primes, then$\ G $ contains a normal Sylow subgroup and is solvable.
I have problem with proving the first part of this (once we prove that there exists such normal Sylow subgroup, we can easily prove by induction that$\ G $ is solvable, using the fact that if$\ N $ is a normal subgroup of$\ G $ such that$\ G/N $ and$\ N $ are solvable, then$\ G $ is solvable).
I know how to prove it for $\ k=2 $ or$\ k=3 $, but I don't have idea how to prove the general case (I didn't manage to do this by induction, also I think that it would require the proving first that$\ G $ is not simple).
I would be grateful for any hints.