Alphabeta MathHow 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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The finite residual is the intersection of the finite-index normal subgroups, and a group is residually finite when that intersection is trivial
Definition
For an abstract group with its profinite-topology data (The profinite topology on a group uses finite-index normal subgroups as an identity-neighbourhood basis, Normal subgroup: invariance under conjugation), the finite residual is
The group is residually finite when . Equivalently, every nonidentity element is omitted by some finite-index normal subgroup.
Depends on
Used by
- The canonical map is injective exactly when the group is residually finite Corollary
- FALSE: the canonical map to the profinite completion is always injective False statement
- Malcev's theorem gives a canonical non-load-bearing source of residually finite groups Remark
- Free groups are residually finite Theorem
- The canonical map to the profinite completion has kernel equal to the finite residual and has dense image Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
- H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)