CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)audited 2026-08-28

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.

C_4 and V_4 can share the same cycle index through nonfaithful actions

Statement refuted

Nonisomorphic groups cannot act on the same finite set with the same cycle index.

Facts & Assumptions

Given: the four-point set $X = {1, 2, 3, 4}$ and the transposition product $τ = (1 2) (3 4)$ .

[F1]

Both $C_{4}$ and $V_{4}$ admit surjective homomorphisms onto the two-element permutation group ${1, τ}$ .

Counterexample

technique · direct

1.1

Use [F1] to let both $C_{4}$ and $V_{4}$ act on $X$ through the same quotient image ${1, τ}$ . In each action, exactly two group elements act as the identity and exactly two act as $τ$ .

2.1

Therefore both actions have the same averaged cycle monomials, namely $\frac{1}{4} (2 s_{1}^{4} + 2 s_{2}^{2}) = \frac{1}{2} (s_{1}^{4} + s_{2}^{2})$ . But $C_{4}$ and $V_{4}$ are not isomorphic. Hence the displayed claim is false.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.