Alphabeta Math
RemarkRemark: Literature-sourcedProof: Not supplied‡ not proved here
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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Malcev's theorem gives a canonical non-load-bearing source of residually finite groups

Statement

Every finitely generated linear group is residually finite.

Remarks

This is the classical Malcev theorem. On this page it is recorded only as a source-backed class of examples and is never load-bearing for a later batch-1 proof.

The local route fails because the theorem is proved by passing from a finitely generated matrix group to a finitely generated coefficient ring and then separating a nontrivial matrix entry modulo a suitable finite quotient of that ring. That commutative-algebra machinery lies outside the present pair.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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