Nilpotence-forcing number

Symbol-free definition
A natural number is said to be nilpotence-forcing if the following equivalent conditions hold:


 * Every group of that order is nilpotent
 * Every group of that order is a direct product of its Sylow subgroups
 * Every prime divisor of that number is Sylow-direct
 * Every prime divisor of that number is Sylow-unique

Stronger properties

 * Abelianness-forcing number
 * Cyclicity-forcing number

Weaker properties

 * Solvability-forcing number