# Difference between revisions of "Cyclicity-forcing number"

From Groupprops

(→Definition) |
(→Definition) |
||

Line 2: | Line 2: | ||

==Definition== | ==Definition== | ||

− | A natural number is termed a '''cyclicity-forcing number''' if it satisfies the following equivalent conditions: | + | A natural number is termed a '''cyclicity-forcing number''' or '''cyclic number''' ([[Wikipedia:cyclic number (group theory)|Wikipedia]]) if it satisfies the following equivalent conditions: |

# There exists exactly one isomorphism class of groups of that [[defining ingredient::order of a group|order]]. | # There exists exactly one isomorphism class of groups of that [[defining ingredient::order of a group|order]]. | ||

Line 8: | Line 8: | ||

# Every group of that order is a direct product of cyclic Sylow subgroups. | # Every group of that order is a direct product of cyclic Sylow subgroups. | ||

# It is a product of distinct primes <math>p_i</math> where <math>p_i</math> does not divide <math>p_j - 1</math> for any two prime divisors <math>p_i, p_j</math> of the order. | # It is a product of distinct primes <math>p_i</math> where <math>p_i</math> does not divide <math>p_j - 1</math> for any two prime divisors <math>p_i, p_j</math> of the order. | ||

+ | # It is relatively prime to its [[defining ingredient:Euler totient function]]. | ||

# It is both a [[defining ingredient::square-free number]] and an [[defining ingredient::abelianness-forcing number]]. | # It is both a [[defining ingredient::square-free number]] and an [[defining ingredient::abelianness-forcing number]]. | ||

# It is both a [[square-free number]] and a [[defining ingredient::nilpotency-forcing number]]. | # It is both a [[square-free number]] and a [[defining ingredient::nilpotency-forcing number]]. |

## Revision as of 01:40, 25 July 2017

*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 defining ingredient: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
- Nilpotence-forcing number
- Solvability-forcing number