DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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The upper central series

Definition

The upper central series of a group $G$ begins with $Z_{0} (G) = 1$ . Having defined $Z_{r} (G) ⊴ G$ , define $Z_{r + 1} (G)$ to be the inverse image of the center $Z (G / Z_{r} (G))$ under the quotient map $G \to G / Z_{r} (G)$ (The center $Z (G)$ of a group, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ). Equivalently, $Z_{r + 1} (G) / Z_{r} (G) = Z (G / Z_{r} (G)) .$ The correspondence theorem makes this inverse image a uniquely determined normal subgroup containing $Z_{r} (G)$ (Correspondence theorem: subgroups of $G / N$ correspond to subgroups of $G$ containing $N$ , with normality preserved). In particular, $Z_{1} (G) = Z (G)$ .

Depends on

Used by

Nilpotent groups and nilpotency class Definition
The dihedral group of order eight is nilpotent of class two Example
Every finite p-group is nilpotent Theorem
Nilpotence via central series, the upper central series, and the lower central series Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)