Simple-feasible number

This article defines a property that can be evaluated for natural numbers

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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 $N$ 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.

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:

• 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 $p^{a}q^{b}$-theorem.