# Cyclicity-forcing number

From Groupprops

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

## Contents

## Definition

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.

## Relation with other properties

### Stronger properties

### Weaker properties

- square-free number
- odd number (except for the special case of the number )
- abelianness-forcing number
- nilpotency-forcing number
- solvability-forcing number