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 $1\leq l\leq k_{j}$.

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

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].