A finite group is nilpotent if and only if all Sylow subgroups are normal, if and only if it is their internal direct product

Given: The hypotheses and objects in the Statement.

Every proper subgroup of a finite nilpotent group is properly contained in its normalizer. (Every proper subgroup of a finite nilpotent group is properly contained in its normalizer).

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. If $N_G(P)\le H\le G$, then $N_G(H)=H$. In particular, $N_G(N_G(P))=N_G(P)$. (A subgroup containing the normalizer of a Sylow subgroup is self-normalizing).

Let $G$ be finite, let $p$ be prime, and write $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $p\nmid m$. Then $G$ has a subgroup of order $p^a$, hence a Sylow $p$-subgroup (def-sylow-p-subgroup). (Sylow I: every finite group has a Sylow $p$-subgroup).

Normal Sylow subgroups for distinct primes centralize one another. (Distinct normal Sylow subgroups centralize one another).

Let $N_0,\dots ,N_{r-1}⊴G$. The following are equivalent: the $N_i$ form an internal direct product of $G$; every $g\in G$ has a unique expression $g=n_0\cdots n_{r-1}$ with $n_i\in N_i$; and the multiplication map $\mu :{\prod }_{i<r}{N}_i\to G$ is an isomorphism. These statements include the empty family and the one-factor case. (Internal direct products are external direct products, equivalently every element has a unique factorisation).

Every subgroup and every quotient of a nilpotent group is nilpotent. Every finite direct product of nilpotent groups is nilpotent; the class of a subgroup or quotient is at most the class of the original group, and the class of a nonempty finite product is at most the maximum of the factor classes. The empty product is the trivial group of class zero. (Subgroups, quotients, and finite direct products of nilpotent groups are nilpotent).

Every finite $p$-group is nilpotent. The trivial group is included and has nilpotency class zero. (Every finite $p$-group is nilpotent).