= Finite nilpotent group decomposition
{title2=$G\cong\prod_{p\mid |G|}P_p$}
A finite group is nilpotent exactly when it is the <direct product of groups> of its <Sylow subgroups>. The <normalizer condition for nilpotent groups> and Sylow conjugacy force each Sylow normalizer to be the whole group. Distinct normal Sylow subgroups commute because their commutators lie in their trivial intersection. Conversely finite p-groups are nilpotent by the nontrivial center and induction; direct products preserve nilpotence.
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