Cyclicity-forcing number

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

Definition

A natural number is termed a cyclicity-forcing number if it satisfies the following equivalent conditions:

1. There exists exactly one isomorphism class of groups of that order.
2. Every group of that order is a cyclic group.
3. Every group of that order is a direct product of cyclic Sylow subgroups.
4. It is a product of distinct primes $p_{i}$ where $p_{i}$ does not divide $p_{j}-1$ for any two prime divisors $p_{i},p_{j}$ of the order.

Equivalence of definitions

For full proof, refer: Classification of cyclicity-forcing numbers

The equivalence of definitions (1)-(3) is not very hard, while the equivalence with part (4) requires some work.

Relation with other properties

Stronger properties

• Prime number

Weaker properties

• Square-free number
• Odd number (except for the special case of the number $2$)
• Abelianness-forcing number
• Nilpotence-forcing number
• Solvability-forcing number