Sylow and maximal-subgroup characterizations of finite nilpotence

Given: The hypotheses and objects in the Statement.

For a finite group $G$, the following are equivalent: $G$ is nilpotent; every Sylow subgroup is normal; and $G$ is the internal direct product of its Sylow subgroups. (A finite group is nilpotent if and only if all Sylow subgroups are normal, if and only if it is their internal direct product).

Every maximal proper subgroup of a finite nilpotent group is normal and has prime index. (Maximal subgroups of finite nilpotent groups are normal of prime index).

Let $G$ be finite and let $\mathrm{\Phi }(G)\le N⊴G$. Then $N$ is nilpotent if and only if $N\mathrm{/}\mathrm{\Phi }(G)$ is nilpotent. In particular, $G$ is nilpotent if and only if $G\mathrm{/}\mathrm{\Phi }(G)$ is nilpotent. (Nilpotence lifts over the Frattini subgroup of a finite group).

For a finite group $G$, the Frattini subgroup is $\mathrm{\Phi }(G):=\bigcap \{M\le G:M\text{ is maximal proper}\}\mathrm{.}$ If $G=1$, the family is empty and its intersection inside $G$ is $G$ itself. Thus $\mathrm{\Phi }(1)=1$. (The Frattini subgroup $\mathrm{\Phi }(G)$ as the intersection of the maximal subgroups of a finite group).

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).

If a prime $p$ divides the order of a finite group $H$, then $H$ contains an element, and hence a subgroup, of order $p$. (Cauchy's theorem: if a prime $p$ divides $\mathrm{\mid }G\mathrm{\mid }$, then $G$ has an element of order $p$).

Every finite $p$-group is nilpotent, including the trivial group. (Every finite $p$-group is nilpotent).

Every subgroup and every quotient of a nilpotent group is nilpotent, and every finite direct product of nilpotent groups is nilpotent. (Subgroups, quotients, and finite direct products of nilpotent groups are nilpotent).

For $K⊴G$, subgroups of $G\mathrm{/}K$ correspond to subgroups of $G$ containing $K$, and the correspondence preserves inclusion and normality. (Correspondence theorem: subgroups of $G\mathrm{/}N$ correspond to subgroups of $G$ containing $N$, with normality preserved).