Nilpotency-forcing number: Difference between revisions

Latest revision as of 14:38, 23 June 2024

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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Nilpotency-forcing number, all facts related to Nilpotency-forcing number) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

A natural number $n$ is said to be nilpotency-forcing or nilpotence-forcing if the following equivalent conditions hold:

Every group of order $n$ is nilpotent
Every group of order $n$ is a direct product of its Sylow subgroups
Every prime divisor of $n$ is Sylow-direct
Every prime divisor of $n$ is Sylow-unique
Suppose $p_{i}, p_{j}$ are prime divisors of $n$ and $p_{j}^{k_{j}}$ is the largest power of $p_{j}$ dividing $n$ . Then, the order of $p_{j}$ modulo $p_{i}$ exceeds $k_{j}$ . In other words, $p_{i}$ does not divide $p_{j}^{l} - 1$ for $1 \leq l \leq k_{j}$ .

For proof of the equivalence of definitions, see classification of nilpotency-forcing numbers.

Relation with other properties

Stronger properties

Weaker properties

Solvability-forcing number

List

The following is a list of all nilpotency-forcing numbers below 100: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99. This sequence is A056867 in the OEIS[1].

@@ Line 1: / Line 1: @@
 {{natural number property}}
+{{wikilocal}}
 ==Definition==
-===Symbol-free definition===
+A natural number <math>n</math> is said to be '''nilpotency-forcing''' or '''nilpotence-forcing''' if the following equivalent conditions hold:
-A natural number is said to be nilpotence-forcing if the following equivalent conditions hold:
+# Every group of order <math>n</math> is [[nilpotent group|nilpotent]]
+# Every group of order <math>n</math> is a direct product of its Sylow subgroups
+# Every prime divisor of <math>n</math> is [[Sylow-direct prime divisor|Sylow-direct]]
+# Every prime divisor of <math>n</math> is [[Sylow-unique prime divisor|Sylow-unique]]
+# Suppose <math>p_i, p_j</math> are prime divisors of <math>n</math> and <math>p_j^{k_j}</math> is the largest power of <math>p_j</math> dividing <math>n</math>. Then, the order of <math>p_j</math> modulo <math>p_i</math> exceeds <math>k_j</math>. In other words, <math>p_i</math> does not divide <math>p_j^l - 1</math> for <math>1 \le l \le k_j</math>.
-* Every group of that order is [[nilpotent group|nilpotent]]
+For proof of the equivalence of definitions, see [[classification of nilpotency-forcing numbers]].
-* Every group of that order is a direct product of its Sylow subgroups
-* Every prime divisor of that number is [[Sylow-direct prime divisor|Sylow-direct]]
-* Every prime divisor of that number is [[Sylow-unique prime divisor|Sylow-unique]]
 ==Relation with other properties==
@@ Line 22: / Line 24: @@
 * [[Solvability-forcing number]]
+==List==
+The following is a list of all nilpotency-forcing numbers below 100: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99.
+This sequence is A056867 in the OEIS[https://oeis.org/A056867].