Nilpotent group (reviewed)

A finite group $G$ is nilpotent if it is a direct product of $p$-groups. Equivalently, all Sylow subgroups of $G$ are normal. Equivalently, $G$ has a central series \[\{1\}=G_0 \lhd G_1 \lhd G_2 \lhd \ldots \lhd G_k = G,\] a nested sequence of normal subgroups with $[G,G_{i+1}] \leq G_i$.

The smallest such $k$ is the nilpotency class of $G$. It is $0$ for $C_1$, $1$ for non-trivial abelian groups, and $\geq 2$ for all other nilpotent groups. By convention, one can define the nilpotency class of a non-nilpotent group to be $-1$.

Nilpotent groups are closed under subgroups and quotients (but not extensions), and are solvable, supersolvable and monomial.