Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01
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.

Every left coset of a subgroup is itself a subgroup

Statement

Every left coset gHgH of every subgroup H≤GH\le G is a subgroup of GG.

Facts & Assumptions

Given: The group S3S_3, the subgroup H={e,(12)}H=\{e,(12)\} and the element g=(123)g=(123) from A left coset that is not the corresponding right coset in Sym⁡({1,2,3})\operatorname{Sym}(\{1,2,3\}).

[F1]

The left coset is gH={(123),(13)}gH=\{(123),(13)\} (Left and right cosets gHgH and HgHg of a subgroup, A left coset that is not the corresponding right coset in Sym⁡({1,2,3})\operatorname{Sym}(\{1,2,3\})).

[F2]

Every subgroup contains the identity element (Subgroup).

Refutation

technique · direct
1.1

Neither (123)(123) nor (13)(13) is the identity, so e∉gHe\notin gH.

F1
2.1

By [F2], the set gHgH cannot be a subgroup. This single coset refutes the statement.

step 1.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 11 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