Equivalence of definitions of finite nilpotent group

Statement
The following are equivalent for a finite group:


 * 1) It is a nilpotent group
 * 2) It satisfies the normalizer condition i.e. it has no proper self-normalizing subgroup
 * 3) Every maximal subgroup is normal
 * 4) All its Sylow subgroups are normal
 * 5) It is the direct product of its Sylow subgroups

Related facts

 * Equivalence of definitions of finite nilpotent Moufang loop
 * Finite ring is internal direct product of its Sylow subrings