Nilpotency-forcing number: Difference between revisions

Revision as of 20:17, 20 June 2016

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

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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

@@ Line 4: / Line 4: @@
 ==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==