Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.
You may only assume the Feit-Thompson theorem here and prove in the following way:
(1) $n$ is a product of odd prime numbers and squarefree.
(2) Then $G$ is solvable. Show that it has a cyclic quotient of prime order, that is there is a an epimorphism $G\to H$with $H$ cyclic of prime order. Let $N$ be the kernel. (Hint: using composition series)
(3)Show that $G\cong N \times H$ and then prove $G$ is abelian.
(4)Show that $G$ is cyclic.
I have proved (1) but get stuck at step 2. Is there any help? Thanks.