Cyclicity-forcing number: Difference between revisions

Latest revision as of 11:02, 22 October 2023

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:

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 $p_{i}$ where $p_{i}$ does not divide $p_{j}-1$ for any two prime divisors $p_{i},p_{j}$ 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

prime number

Weaker properties

square-free number
odd number (except for the special case of the number $2$ )
abelianness-forcing number
nilpotency-forcing number
solvability-forcing number

List

The following is a list of all cyclicity-forcing numbers below 100: 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97. The non-prime numbers are highlighted in bold.

This sequence is A003277 in the OEIS[1].

@@ Line 2: / Line 2: @@
 ==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]].
@@ Line 8: / Line 8: @@
 # 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 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 [[square-free number]] and a [[defining ingredient::nilpotency-forcing number]].
 ===Equivalence of definitions===
-{{proofat|[[Classification of cyclicity-forcing numbers]]}}
+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.
-The equivalence of definitions (1)-(3) is not very hard, while the equivalence with part (4) requires some work.
 ==Relation with other properties==
 ===Stronger properties===
-* [[Weaker than::Prime number]]
+* [[Weaker than::prime number]]
 ===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::nilpotency-forcing number]]
-* [[Stronger than::Solvability-forcing number]]
+* [[Stronger than::solvability-forcing number]]
+==List==
+The following is a list of all cyclicity-forcing numbers below 100: '''1''', 2, 3, 5, 7, 11, 13, '''15''', 17, 19, 23, 29, 31, '''33''', '''35''', 37, 41, 43, 47, '''51''', 53, 59, 61, '''65''', 67, '''69''', 71, 73, '''77''', 79, 83, '''85''', '''87''', 89, '''91''', '''95''', 97. The non-prime numbers are highlighted in bold.
+This sequence is A003277 in the OEIS[https://oeis.org/A003277].