Abelianness-forcing number

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

Definition

A natural number $n$ is said to be an abelian number or 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 $k_{i}\leq 2$ for all $i$ AND $p_{i}$ does not divide $p_{j}^{k_{j}}-1$ for any $1\leq i,j\leq r$
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
prime number	every group of that order is cyclic	follows from cyclic implies abelian

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

List

The following is a list of all abelian-forcing numbers below 100: 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99. This sequence is A051532 in the OEIS[1].