Simple-feasible number

Symbol-free definition
A natural number is said to be simple-feasible if there is a simple group with that natural number as its order.

Definition with symbols
A natural number $$N$$ is said to be simple-feasible if there is a simple group $$G$$ whose order is $$N$$.

For simplicity, we shall assume that $$N$$ is a composite number (because for $$N$$ prime we anyway know that there is a unique simple group of that order.

Direct facts from Sylow theory
For a number to be simple-feasible, one must be ensured of the nonexistence of normal subgroups for at least one group of that order. Thus, presence of the following kinds of divisors immediately rules out simple-feasibility:


 * Sylow-unique prime divisor
 * Core-nontrivial prime divisor
 * Closure-proper prime divisor

Some related facts:


 * Prime divisor greater than Sylow index is Sylow-unique
 * Order is product of Mersenne prime and one more implies normal Sylow subgroup
 * Order of simple non-Abelian group divides factorial of every Sylow number

Other facts

 * A5 is the simple non-Abelian group of smallest order: In particular, this shows that no composite number less than $$60$$ is simple-feasible.
 * Odd-order implies solvable: Any group of odd order is solvable. In particular, no odd composite number is simple-feasible. This result is termed the odd-order theorem and also the Feit-Thompson theorem.
 * Prime power order implies nilpotent: Any group whose order is a power of a prime is nilpotent. In particular, no prime power, other than a prime itself, is simple-feasible.
 * Order has only two prime factors implies solvable: Any group whose order has only two prime factors is solvable. This is called Burnside's $$p^aq^b$$-theorem.