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 or cyclic number (Wikipedia) 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.
5. It is relatively prime to its Euler totient function.
6. It is both a square-free number and an abelianness-forcing number.
7. It is both a square-free number and a nilpotency-forcing number.

Equivalence of definitions

The equivalence of definitions (1)-(3) is not very hard, while the equivalence with part (4) is covered by the classification of cyclicity-forcing numbers. We can also demonstrate the equivalence with (5) and (6), by combining with the classification of abelianness-forcing numbers and classification of nilpotency-forcing numbers respectively.

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
• nilpotency-forcing number
• solvability-forcing number