Finite nilpotent group

Main equivalent definitions
A finite group is termed a finite nilpotent group if it satisfies the following equivalent conditions:


 * 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
 * 6) It is a p-nilpotent group for every prime number $$p$$ (it suffices to check this condition only for those primes that divide the order). $$p$$-nilpotent means that there exists a normal p-complement.
 * 7) It has a normal subgroup for every possible order dividing the group order
 * 8) Every normal subgroup of the group contains a normal subgroup of the group for every order dividing the order of the normal subgroup.

Other equivalent definitions that are weaker versions of nilpotent in the general case
The following is a list of group properties, each weaker than being nilpotent, that for a finite group turn out to be equivalent to being nilpotent:


 * Group satisfying normalizer condition: It has no proper self-normalizing subgroup
 * Group in which every subgroup is subnormal
 * Locally nilpotent group
 * Residually nilpotent group
 * Engel group: See Zorn's theorem on Engel groups
 * Hypercentral group
 * Hypocentral group