Simple-feasible number

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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 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