This article defines a property that can be evaluated for natural numbers
A natural number is termed a cyclicity-forcing number or cyclic number (Wikipedia) if it satisfies the following equivalent conditions:
- There exists exactly one isomorphism class of groups of that order.
- Every group of that order is a cyclic group.
- Every group of that order is a direct product of cyclic Sylow subgroups.
- It is a product of distinct primes where does not divide for any two prime divisors of the order.
- It is relatively prime to its Euler totient function.
- It is both a square-free number and an abelianness-forcing number.
- 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.