Prove that if a finite solvable group is simple, it is a cyclic group of prime order.

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Prove that if a finite solvable group is simple, it is a cyclic group of prime order.

Help me some hints.

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Hints:

  • $[G, G] \triangleleft G$, and
  • $[G, G] \neq G$ for a solvable group $G \neq 1$ (why?).