Nilpotency-forcing number

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.