# Nilpotent of cube-free order implies abelian

From Groupprops

## Statement

Suppose is a Finite nilpotent group (?) and there is no prime number such that divides the order of . In other words, the order of is a cube-free number.

Then, is a Finite abelian group (?).

## Related facts

## Facts used

- Classification of groups of prime-square order, showing that any group of prime-square order is abelian.
- Equivalence of definitions of finite nilpotent group, in particular the part that states that any nilpotent group is the internal direct product of its Sylow subgroups.