# nLab upper central series

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# Contents

## Idea

The upper central series of a group, $G$, is an ascending sequence of subgroups

$Z^0(G) \leq Z^1(G) \leq Z^2(G) \leq \ldots,$

where the zeroth member is the trivial group, the first member is the center, the second member is the second center, and so on.

$Z^0(G) \,=\, 1, \;\; Z^1(G) = Z(G), \;\; Z^2(G)/Z^1(G) = Z(G/Z^1(G)), \;\; Z^3(G)/Z^2(G) = Z(G/Z^2(G)), \;\; \ldots$

The indexing may be continued to the ordinals, where the member indexed by a limiting ordinal is the union of all previous members.

For a nilpotent group, the upper central series reaches the whole group in finitely many steps, and is the fastest ascending central series. It has the same length then as the lower central series, although they need not coincide.