On the other hand, Feit-Thompson's theorem required 255 pages of heavy machinery to prove.

Oh, that puts a different complexion on the matter then.

]]>There. That was not really heavy machinery, was it?

That solvability of composite order implies nonsimplicity is not, no. In fact, it becomes more obvious when you use the decomposition series that comes from being solvable. On the other hand, Feit-Thompson's theorem required 255 pages of heavy machinery to prove. I'm not sure of any proofs discovered since then, so perhaps it has become shorter with time.

]]>And this implies that all finite groups of odd composite order are not simple? I can prove it.

Let be a finite group of composite odd order (so its soluble by FeitThompson). We may assume that is not Abelian, since we all know that any Abelian group of composite order is not simple. Since it is not Abelian, its derived subgroup is not trvial. Also cannot be all of otherwise for all whereas being soluble means that must be trivial after a finite number of derivations. Hence the commutator subgroup of is a nontrivial and proper normal subgroup, proving that is not simple.

There. That was not really heavy machinery, was it?

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