"Prove there does not exist a finite simple non-abelian group of order of a Fibonacci number"

I would like to answer the above question, but I currently have few ideas of where to begin.

I understand we will likely only be using results about the prime factorisation of fibonacci numbers. I considered basic results about simple groups, e.g. from the sylow theorems, if a prime $p | |G| $ and $ kp+1$ does not divide $|G|$ for all integers $k$, then the sylow-p subgroup is normal.

However I fail to see how exactly to use this, and other standard techniques.

I have heard before that no Fibonacci number is a perfect number, but again, I cannot see how to use this exactly.

**Would someone be able to provide me with hints/ideas?**

**In particular, is there a specific property of simple groups or Fibonacci numbers that I need consider?**