Nilpotency-forcing number

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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]

Definition

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

  1. Every group of order n is nilpotent
  2. Every group of order n is a direct product of its Sylow subgroups
  3. Every prime divisor of n is Sylow-direct
  4. Every prime divisor of n is Sylow-unique
  5. 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 \le l \le k_j.

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

Relation with other properties

Stronger properties

Weaker properties