False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

FALSE: any transposition together with any $n$ -cycle generates $S_{n}$

Statement refuted

Every $n$ -cycle and every transposition together generate $S_{n}$ .

Facts & Assumptions

Given: The elements $c = (1234)$ and $t = (13)$ of $S_{4}$ .

[F1]

Relative to $(1 2 \dots n)$ , the neighboring transposition $(1 2)$ does generate $S_{n}$ with that cycle (For $n \geq 2$ , $(1 2 \dots n)$ and $(1 2)$ generate $S_{n}$ ).

[F2]

A generated subgroup is the smallest subgroup containing its generators (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[F3]

Permutations act on the underlying set by composition (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition).

Counterexample

technique · counterexample

1.1

In $S_{4}$ , let $c = (1234)$ and $t = (13)$ , and partition the symbols into $B_{1} = {1, 3}$ and $B_{2} = {2, 4}$ .

2.1

The cycle $c$ swaps $B_{1}$ and $B_{2}$ , while $t$ preserves each block. Therefore every word in $c, t$ preserves the two-block system setwise.

F2F3step 1.1

3.1

The permutation $(12)$ does not preserve that block system, so it is not in $⟨ c, t ⟩$ ; hence this generated subgroup is proper in $S_{4}$ .

F2step 2.1

4.1

Thus an arbitrary transposition need not work. The positive theorem [F1] requires a neighboring transposition relative to the chosen cycle.

F1step 3.1∎

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: 20 results over 15 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

M. Macauley, Math 4120 lecture notes: Generating sets for $S_n$ (standard reference, not scraped)

FALSE: any transposition together with any nn-cycle generates SnSn​