# Central series of a group

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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.

#### Comments

How to Cite This Entry:
Central series of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_series_of_a_group&oldid=44590
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article