Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01
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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.

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Every left coset of a subgroup is itself a subgroup

Statement

Every left coset gH of every subgroup H≤G is a subgroup of G.

Facts & Assumptions

Given: The group S3, the subgroup H={e,(12)} and the element g=(123) from A left coset that is not the corresponding right coset in Sym⁡({1,2,3}).

[F1]

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

[F2]

Every subgroup contains the identity element (Subgroup).

Refutation

technique · direct
1.1

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

F1
2.1

By [F2], the set gH 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 · two levels

9 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