# Central series of a group

A normal series all factors of which are central, that is, a series of subgroups

$$E=G_0\subseteq G_1\subseteq\ldots\subseteq G_n=G$$

for which $G_{i+1}/G_i$ lies in the centre of $G/G_i$ for all $i$ (see also Subgroup series). If for all $i$ the subgroup $G_{i+1}/G_i$ is the complete centre of $G/G_i$, then the series is called the upper central series of $G$ and if the commutator subgroup of $G_{i+1}$ and $G$ coincides with $G_i$, then the series is called the lower central series of $G$.

A group having a central series is called a nilpotent group. In a nilpotent group the lower and the upper central series have the same length, which is the minimal length of a central series of the group. This length is called the nilpotency class (or the degree of nilpotency) of the group.