Equivalence of definitions of finite nilpotent group

This article gives a proof/explanation of the equivalence of multiple definitions for the term finite nilpotent group
View a complete list of pages giving proofs of equivalence of definitions

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