Simple-feasible number: Difference between revisions

Revision as of 00:06, 6 January 2009

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]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Simple-feasible number, all facts related to Simple-feasible number) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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:

Some related facts:

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.

@@ Line 31: / Line 31: @@
 * [[Closure-proper prime divisor]]
-===Odd-order theorem===
+Some related facts:
-The [[Feit-Thompson theorem]] tells us that any group of odd order is [[solvable group|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.
+* [[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 <math>60</math> 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.