# Nilpotency-forcing number

From Groupprops

*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 is said to be **nilpotency-forcing** or **nilpotence-forcing** if the following equivalent conditions hold:

- Every group of order is nilpotent
- Every group of order is a direct product of its Sylow subgroups
- Every prime divisor of is Sylow-direct
- Every prime divisor of is Sylow-unique
- Suppose are prime divisors of and is the largest power of dividing . Then, the order of modulo exceeds . In other words, does not divide for .

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