Equivalence of definitions of finite nilpotent group

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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

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