# Solvability-forcing number

From Groupprops

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

## Contents

## Definition

### Symbol-free definition

A natural number is said to be **solvability-forcing** if it satisfies the following equivalent conditions:

- Every group of that order is solvable
- It has no non-prime divisor which is simple-feasible. In other words, no divisor of it occurs as the order of a simple non-abelian group

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

odd number | not divisible by 2 | odd-order implies solvable (also known as the Feit-Thompson theorem or odd-order theorem) | any nontrivial power of 2 offers a counterexample | |FULL LIST, MORE INFO |

a number whose order has at most two distinct prime factors | order has only two prime factors implies solvable (also known as Burnside's p^aq^b theorem | any odd number with three or more distinct prime factors; also, any square-free number with three or more distinct prime factors | ||

square-free number | a number whose order is a product of distinct primes | Square-free implies solvability-forcing | any prime power, such as the square of a prime | |FULL LIST, MORE INFO |

nilpotency-forcing number | a number such that any group of that order is nilpotent | follows from nilpotent implies solvable | a number such as 6 is solvability-forcing but not nilpotency-forcing | |FULL LIST, MORE INFO |

abelianness-forcing number | a number such that any group of that order is abelian | (via nilpotency-forcing) | (via nilpotency-forcing) | |FULL LIST, MORE INFO |

cyclicity-forcing number | a number such that any group of that order is cyclic | (via abelianness-forcing) | (via abelianness-forcing) | Abelianness-forcing number, Square-free number|FULL LIST, MORE INFO |