Cyclicity-forcing number

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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

Weaker properties