Nilpotency-forcing number

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:

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

• Abelianness-forcing number
• Cyclicity-forcing number

Weaker properties

• Solvability-forcing number