Difference between revisions of "Cyclicity-forcing number"

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(Definition)
(Relation with other properties)
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Prime number]]
+
* [[Weaker than::prime number]]
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Square-free number]]
+
* [[Stronger than::square-free number]]
* [[Stronger than::Odd number]] (except for the special case of the number <math>2</math>)
+
* [[Stronger than::odd number]] (except for the special case of the number <math>2</math>)
* [[Stronger than::Abelianness-forcing number]]
+
* [[Stronger than::abelianness-forcing number]]
* [[Stronger than::Nilpotence-forcing number]]
+
* [[Stronger than::nilpotence-forcing number]]
* [[Stronger than::Solvability-forcing number]]
+
* [[Stronger than::solvability-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 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