Abelianness-forcing number

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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