DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Dependency tree · 2 levels

Sources