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
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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