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

List

The following is a list of all cyclicity-forcing numbers below 100: 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97. The non-prime numbers are highlighted in bold.

This sequence is A003277 in the OEIS[1].