Constrained for a prime divisor implies not simple non-abelian
Further information: P-constrained group (?)
We prove the contrapositive, which is somewhat easier.
Given: A finite simple non-abelian group
To prove: is not -constrained for any prime divisor of the order of .
Proof: Since divides the order of , we obtain that is trivial, and hence is trivial for any -Sylow subgroup of . We thus get:
Since is trivial, we see that the -constraint condition is violated.