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\dotsb\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.