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
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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.
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:
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
- A5 is the simple non-Abelian group of smallest order: In particular, this shows that no composite number less than 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 -theorem.