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.
- 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.
FALSE: the canonical map to the profinite completion is always injective
Statement
The canonical map to the profinite completion is always injective.
Facts & Assumptions
Given: A group that is not residually finite.
The canonical map is injective exactly when the group is residually finite (The canonical map to the profinite completion has kernel equal to the finite residual and has dense image, The finite residual is the intersection of the finite-index normal subgroups, and a group is residually finite when that intersection is trivial).
Refutation
Because is not residually finite, the finite residual contains some nonidentity element.
By [L1], the finite residual is exactly the kernel of the canonical map. Hence that nonidentity element lies in the kernel, so the canonical map is not injective.
Therefore the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
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)