# Abelianness-forcing number

From Groupprops

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

## Contents

## Definition

A natural number is said to be **abelianness-forcing** if the following equivalent conditions hold:

- Every group of order is abelian
- Every group of order is an internal direct product of abelian Sylow subgroups
- has prime factorization of the form with for all
**AND**does not divide for any - is a cube-free number as well as a nilpotency-forcing number

### Equivalence of definitions

The equivalence of (1) and (2) is direct.

The equivalence with (3) follows from the classification of abelianness-forcing numbers.

The equivalence with (4) follows by combining with the classification of nilpotency-forcing numbers.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication 1! Proof of strictness (reverse implication failure) | Intermediate notions | |
---|---|---|---|---|

cyclicity-forcing number | every group of that order is cyclic | follows from cyclic implies abelian | any square of a prime is abelianness-forcing but not cyclicity-forcing |

### Weaker properties

Property | Meaning | Proof of implication 1! Proof of strictness (reverse implication failure) | Intermediate notions | |
---|---|---|---|---|

nilpotency-forcing number | every group of that order is nilpotent | follows from abelian implies nilpotent | |FULL LIST, MORE INFO | |

Solvability-forcing number | every group of that order is solvable | (via nilpotency-forcing) | (via nilpotency-forcing) | |FULL LIST, MORE INFO |