Simple-feasible number
This article defines a property that can be evaluated for natural numbers
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article is about a term related to the Classification of finite simple groups
History
This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page
Definition
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 is said to be simple-feasible if there is a simple group whose order is .
For simplicity, we shall assume that is a composite number (because for prime we anyway know that there is a unique simple group of that order.
Facts
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:
Odd-order theorem
The Feit-Thompson theorem tells us that any group of odd order is solvable. We know that the only simple groups that are solvable are the cyclic groups of prime order. Thus, no odd number (other than a prime) can be simple-feasible.