Difference between revisions of "Cyclicity-forcing number"

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(Definition)
(Definition)
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==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]].
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# 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

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 defining ingredient: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