DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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.

A free basis of a group

Definition

Let $F$ be a group and let $B\subseteq F$ . Write $i:B\hookrightarrow F$ for the inclusion. The subset $B$ is a free basis of $F$ if $(F,i)$ is a free group on the set $B$ in the sense of Free group on a set of generators. Equivalently, for every group $G$ and every function $u:B\to G$ , there is a unique group homomorphism $\widehat u:F\to G$ whose restriction to $B$ is $u$ .

Depends on

Free group on a set of generators

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 8 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

Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.3 (standard reference, not scraped)
Encyclopedia of Mathematics, Free group (standard reference, not scraped)