# Abelianness-forcing number

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

## Definition

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

1. Every group of order $n$ is abelian
2. Every group of order $n$ is an internal direct product of abelian Sylow subgroups
3. $n$ has prime factorization of the form $n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}$ with $k_i \le 2$ for all $i$ AND $p_i$ does not divide $p_j^{k_j} - 1$ for any $1 \le i,j \le r$
4. $n$ 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