Constrained for a prime divisor implies not simple non-abelian
Further information: P-constrained group (?)
Given: A finite group , a prime divisor of the order of . is -constrained.
To prove: is not a simple non-abelian group.
Proof: Since divides the order of , there exists a nontrivial -Sylow subgroup of . By fact (1), is nontrivial, so the centralizer , which contains is nontrivial. Thus, by the definition of -constrained, we have that is nontrivial. Thus, it is a nontrivial normal subgroup of , so we have .
Consider . If is trivial, then is a -group, and hence by fact (1), is a nontrivial normal subgroup, forcing , forcing to be abelian.
On the other hand, if , then does not divide the order of , a contradiction.